SECOND LAW ANALYSIS OF SOLID OXIDE FUEL CELLS A ...

SECOND LAW ANALYSIS OF SOLID OXIDE FUEL CELLS

A THESIS SUBMITTED TO

THE GRADUATE SCHOOL OF NATURAL AND APPLIED SCIENCES

OF

THE MIDDLE EAST TECHNICAL UNIVERSITY

BY

BAŞAR BULUT

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE

DEGREE OF

MASTER OF SCIENCE

IN

THE DEPARTMENT OF MECHANICAL ENGINEERING

SEPTEMBER 2003

iii

ABSTRACT

SECOND LAW ANALYSIS OF SOLID OXIDE FUEL CELLS

Bulut, Başar

M.S., Department of Mechanical Engineering

Supervisor : Assoc. Prof. Dr. Cemil Yamalı

Co-Supervisor : Prof. Dr. Hafit Yüncü

September 2003, 111 pages

In this thesis, fuel cell systems are analysed thermodynamically and electrochemically.

Thermodynamic relations are applied in order to determine the change of first law and

second law efficiencies of the cells, and using the electrochemical relations, the

irreversibilities occuring inside the cell are investigated.

Following this general analysis, two simple solid oxide fuel cell systems are examined.

The first system consists of a solid oxide unit cell with external reformer. The second

law efficiency calculations for the unit cell are carried out at 1273 K and 1073 K, 1 atm

and 5 atm, and by assuming different conversion ratios for methane, hydrogen, and

oxygen in order to investigate the effects of temperature, pressure and conversion ratios

on the second law efficiency. The irreversibilities inside the cell are also calculated and

iv

graphed in order to examine their effects on the actual cell voltage and power density of

the cell.

Following the analysis of a solid oxide unit cell, a simple fuel cell system is modeled.

Exergy balance is applied at every node and component of the system. First law and

second law efficiencies, and exergy loss of the system are calculated.

Keywords: Exergy, Solid oxide fuel cell, Second law efficiency

v

ÖZ

KATI OKSİT YAKIT HÜCRELERİNİN İKİNCİ KANUN ANALİZİ

Bulut, Başar

Yüksek Lisans, Makina Mühendisliği Bölümü

Tez Yöneticisi : Doç. Dr. Cemil Yamalı

Ortak Tez Yöneticisi : Prof. Dr. Hafit Yüncü

Eylül 2003, 111 sayfa

Bu tez çalışmasında, yakıt hücresi sistemleri termodinamiksel ve elektrokimyasal olarak

analiz edilmiştir. Hücrelerin birinci ve ikinci kanun verimlerinin değişiminin

incelenmesi için genel termodinamik bağıntılar kullanılmış, elektrokimyasal bağıntılar

kullanaran hücre içinde meydana gelen tersinmezlikler incelenmiştir.

Bu genel analizin ardından, iki basit katı oksit yakıt hücresi sistemi incelenmiştir. Birinci

sistem dış düzenleyicili bir katı oksit hücresinden oluşmaktadır. İkinci kanun verim

hesaplamaları, sıcaklık, basınç ve değişme oranlarının ikinci kanun verimine etkilerini

inceleyebilmek amacıyla 1273 K ve 1073 K, 1 atm ve 5 atm, ve de metan, hidrojen ve

oksijen için değişik değişme oranları varsayılarak tamamlanmıştır. Hücre içerisindeki

vi

tersinmezlikler de hesaplanmış ve fiili hücre voltajına ve hücrenin güç yoğunluğuna olan

etkilerini inceleyebilmek için grafikleri çizilmiştir.

Katı oksit hücre analizini, basit bir katı oksit sisteminin modellenmesi takip etmiştir.

Ekserji denklemi, sistem içindeki her parça ve düğüm noktasına uygulanmıştır. Birinci

ve ikinci kanun verimleri ve sistemin ekserji kayıpları hesaplanmıştır.

Anahtar Kelimeler: Ekserji, Katı oksit yakıt hücresi, İkinci kanun verimi

vii

To My Parents

viii

ACKNOWLEDGMENTS

I would like to thank to my supervisor Assoc. Prof. Dr. Cemil Yamalı for his guidance

and insight throughout the research. I express sincere appretiation to my co-supervisor

Prof. Dr. Hafit Yüncü for his guidance, suggestions and comments.

I express sincere thanks to my family for their support and faith in me, and for their

understanding in every step of my education.

ix

TABLE OF CONTENTS

ABSTRACT ........................................................................................................... iii

ÖZ ........................................................................................................................... v

ACKNOWLEDGMENTS....................................................................................... viii

TABLE OF CONTENTS ........................................................................................ ix

LIST OF TABLES .................................................................................................. xiii

LIST OF FIGURES................................................................................................. xvi

LIST OF SYMBOLS .............................................................................................. xix

CHAPTER

1. INTRODUCTION................................................................................. 1

1.1 Definition of a Fuel Cell............................................................. 1

1.2 Fuel Cell Plant Description ........................................................ 4

1.3 Fuel Cell Stacking ...................................................................... 4

1.4 Characteristics of Fuel Cells ...................................................... 6

1.4.1 Efficiency ...................................................................... 6

1.4.2 Flexibility in Power Plant Design ................................. 7

1.4.3 Manufacturing and Maintenance................................... 7

1.4.4 Noise.............................................................................. 7

x

1.4.5 Heat................................................................................ 8

1.4.6 Low Emissions .............................................................. 8

1.5 Types of Fuel Cells and Their Fields of Applications................. 9

2. THERMODYNAMICS OF FUEL CELLS .......................................... 11

2.1 Some Fundamental Relations..................................................... 11

2.1.1 TdS Equations and Maxwell Relations ......................... 11

2.1.2 Partial Molal Properties................................................. 16

2.1.3 Chemical Potential......................................................... 18

2.2 Thermodynamics of Chemical Reactions................................... 19

2.2.1 Free Energy Change of Chemical Reactions................. 19

2.2.2 Standard Free Energy Change of a Chemical Reaction. 19

2.2.3 Relation Between Free Energy Change in a Cell Reaction and Cell Potential .......................................................... 20

2.3 Nernst Equation.......................................................................... 22

2.4 Exergy Concept .......................................................................... 24

2.4.1 Exergy Balance.............................................................. 24

2.4.2 Chemical Exergy ........................................................... 26

2.4.3 Physical Exergy............................................................. 27

2.5 Efficiency of Fuel Cells.............................................................. 27

2.5.1 Thermodynamic ( First and Second Law ) Efficiencies 27

2.5.2 Electrochemical Efficiencies......................................... 33

3. KINETIC EFFECTS ............................................................................. 34

3.1 Introduction ................................................................................ 34

3.2 Fuel Cell Irreversibilities............................................................ 37

xi

3.2.1 Activation Polarization.................................................. 38

3.2.2 Ohmic Polarization........................................................ 42

3.2.3 Concentration Polarization ............................................ 42

3.3 Mass Transport Effects............................................................... 46

3.3.1 Knudsen Diffusion......................................................... 46

3.3.2 Molecular Diffusion ...................................................... 48

3.3.3 Transition Region Diffusion.......................................... 48

4. MODELING AND CALCULATION .................................................. 54

4.1 Fuel Cell Type Selection ............................................................ 54

4.2 Environment and Air Composition ............................................ 55

4.3 Chemical Reactions and Components of SOFC System ........... 55

4.4 Simulation Models...................................................................... 58

4.4.1 Simulation Model 1: SOFC Unit Analysis.................... 58

4.4.2 Simulation Model 2: SOFC System Analysis ............... 59

4.5 Electrochemical Model............................................................... 60

4.6 Heat Exchanger Model............................................................... 63

4.7 Calculation Procedure................................................................. 65

4.7.1 General Assumptions..................................................... 66

4.7.2 Calculation Steps for Simulation Model 1..................... 67

4.7.3 Calculation Steps for Simulation Model 2 .................... 69

5. RESULTS AND DISCUSSION............................................................ 72

5.1 Results of Simulation Model 1................................................... 72

5.1.1 Electrochemical Model Analysis................................... 72

5.1.2 Thermodynamic Analysis.............................................. 83

xii

5.2 Results of Simulation Model 2................................................... 98

5.2.1 The Heat Required by The Reformer and Vaporizer .... 98

5.2.2 Heat Exchanger Design .................................................101

5.2.3 Thermodynamic Analysis..............................................102

6. CONCLUSION......................................................................................108

REFERENCES........................................................................................................110

LIST OF TABLES

TABLE

1.1 Classification of fuel cells ................................................................... 9

1.2 Application fields of fuel cells............................................................. 10

4.1 Properties of SOFC materials ............................................................... 55

5.1 Mixture compositions for ideal case ( i.e. 14=CHη , 1

2=Hη ,

12=Oη ) ……………………………………………………………… 84

5.2 Mixture compositions for 9.04=CHη , 9.0

2=Hη , 9.0

2=Oη ……….. 84


2=Hη , 8.0

2=Oη ……….. 84


2=Hη , 7.0

2=Oη ……….. 85


2=Hη , 8.0

2=Oη ……… 85


2=Hη , 7.0

2=Oη ……….. 85


2=Hη , 7.0

2=Oη ………... 86

5.8 Calculated second law efficiencies as functions of conversion ratios ( T = 1273 K, P = 1 atm, I = 8000 A / m² ) ……………………. 87

5.9 Calculated second law efficiencies as functions of conversion

ratios ( T = 1073 K, P = 1 atm, I = 8000 A / m² )…………………….87

xiii

xiv

5.10 Calculated second law efficiencies as functions of conversion

ratios ( T = 1273 K, P = 1 atm, I = 6000 A / m² ) …………………. 88

5.11 Calculated second law efficiencies as functions of conversion ratios ( T = 1073 K, P = 1 atm, I = 6000 A / m² ) …………………. 88













5.24 Heat exchanger design conditions and results ……………………...102

5.25 Calculated molar chemical compositions of gas streams at each node……………………………………………………………103

xv

5.26 Net work output of the SOFC operating at Ta = 1000 K, Tc = 900 K, I = 1000 A/m² …………………………………..……104

5.27 The first and second law efficiencies of model 2 …………………...104

5.28 The comparison of input, output, and loss of energy and exergy in the system. Energy and exergy values are normalized relative to the lower heating value and chemical exergy of the fuel, respectively ……………………………………………………106

xvi

LIST OF FIGURES

FIGURES

1.1 Schematic of an individual fuel cell................................................. ..... 2

1.2 Direct energy conversion with fuel cells in comparison to

to indirect energy conversion ............................................................... 3

1.3 Fuel cell power plant processes .......................................................... 5

1.4 Stacking of individual fuel cells ......................................................... 6

2.1 An open thermodynamic system with single inlet and outlet ............. 24

2.2 Simple H2/O2 fuel cell system, T = 25°C ............................................ 28

2.3 Schematic of the system used to calculate the second law efficiency of the simple fuel cell ………………………………….... 30

2.4 Comparison between fuel cell first law and second law efficiency changes with temperature and Carnot efficiency change with temperature……………………………………………… 32

3.1 Voltage change with current density for a simple fuel cell operating at about 40°C, and at standard pressure ................................ 35

3.2 Voltage change with current density for a solid oxide fuel cell operating at about 800°C ................................................................ 36

3.3 The film thickness theory .................................................................... 43

3.4 Types of diffusion through the pores: ( a ) Knudsen diffusion, ( b ) molecular diffusion, ( c ) transition diffusion................................ 47

xvii

4.1 Schematic of simulation model 1 ........................................................ 58

4.2 Schematic of simulation model 2 ........................................................ 59

5.1 Activation polarization change with current density ( T = 1273 K )…………………………………………………………. 73

5.2 Ohmic polarization change with current density ( T = 1273 K ) …… 73

5.3 Concentration polarization change with current density ( T = 1273 K )…………………………………………………………. 74

5.4 Change in the cell voltage and the power density with current density ( T = 1273 K )………………………………………… 74

5.5 Calculated polarization effects with current density ( T = 1273 K )…………………………………………………………. 75

5.6 Calculated power density, cell voltage and polarizations with current density ( 1273 K )…..…………………………………… 76

5.7 Activation polarization change with current density ( T = 1073 K ) ………………………………………………………. 77

5.8 Ohmic polarization change with current density ( T = 1073 K ) ..….... 78

5.9 Concentration polarization change with current density ( T = 1073 K )…………………………………………………………. 78

5.10 Change in the cell voltage and the power density with current density ( T = 1073 K )……………………………………… 79

5.11 Calculated polarization effects with current density ( T = 1073 K )………………………………………………………. 80

5.12 Calculated power density, cell voltage and polarizations with current density ( 1073 K )……………………………………… 81

5.13 Second law efficiency with current density at P = 1 atm, and P = 5 atm, conversion ratios are 100% and 90% respectively, T = 1273 K …………………………………………………………. 95

5.14 Second law efficiency with current density at P = 1 atm, and P = 5 atm, conversion ratios are 100% and 90% respectively, T = 1073 K …………………………………………………………. 96

xviii

5.15 Second law efficiency with current density at T = 1273 K, and T = 1073 K, conversion ratios are 100% and 90% respectively, P = 1 atm……………………………………………………………. 96

5.16 Second law efficiency with current density at T = 1273 K, and T = 1073 K, conversion ratios are 100% and 90% respectively, P = 5 atm……………………………………………………………. 97

5.17 Heat requirement of the components of the SOFC system with methane reforming rate ( Tr,i = 1100 K )…………………………… 99

5.18 Heat release of the combustion processes in the afterburner With respect to the mole number of the fuel that is burnt………….. 99

5.19 Comparison of heat release by methane and hydrogen with reformer efficiency …………………………………………………100

5.20 Change in fuel utilization rate with reformer efficiency …………...101

5.21 System 2 operating at 1 atm, 90% reformer efficiency, 75% fuel utilization rate, fuel inlet temperature is 1000 K, air inlet temperature is 900 ………………………………………………......105

xix

LIST OF SYMBOLS

a Activity C Concentration ( mole / m³ ) D Diffusion coefficient ( m² / s ) e Energy of molecular interaction ( ergs ) E Cell voltage F Faraday constant ( = 96487 kJ / V.kmole electrons ) G Gibbs free energy ( jJ ) i Current density ( A / m² ) io Exchange current density ( A / m² ) I Current density ( A / m² ) IL Limiting current density ( A / m² ) İ Irreversibility ( kJ / s ) J Mass flux ( kg / s ) K Equilibrium constant n Number of moles ne Electrons transferred per reaction M Molecular mass p Partial pressure ( atm ) P Pressure ( atm ) Q Heat ( kJ ) R Universal gas constant ( = 8.3145 kJ / kmole K ) Re Area specific resistance ( Ω / m² ) T Temperature ( K ) w Thickness ( µm ) W Work ( kJ ) We Electrical work ( kJ ) X Mole fraction

xx

Greek Letters α Transfer coefficient δ Thickness of the diffusion layer ( m ) ε Porosity η Polarization ( V ) ηI First law efficiency ηII Second law efficiency µ Chemical potential ( kJ / mole ) ρ Resistivity ( Ω cm ) ξ Tortuosity σ Collision diameter ( Ả ) ΩD Collision integral nased on the Lennard-Jones potential Subscripts a Anode A A specie B B specie c Cathode k Knudsen diffusion P Products R Reactants rev Reversible (eff) Effective Superscripts I Inlet condition

1

CHAPTER 1

INTRODUCTION

1.1 Definition of a Fuel Cell

A fuel cell is an electrochemical device which can continuously convert the free energy

of the reactants ( i.e. the fuel and the oxidant ), which are stored outside the cell itself,

directly to electrical energy. The basic physical structure or building block of a fuel cell

consists of an electrolyte layer in contact with two porous electrodes; the anode or the

fuel electrode, where the fuel that feeds the cell is oxidised, and the cathode or the

oxygen ( or air ) electrode, where the reduction of molecular oxygen occurs, on either

side. A schematic representation of a fuel cell with the reactant / product gases and the

ion conduction flow directions through the cell is shown in Figure 1.1 [1].

In a fuel cell, gaseous fuels are fed continuously to the anode ( negative electrode ) and

an oxidant ( i.e. oxygen from air ) is fed continuously to the cathode ( positive electrode

). The electrochemical reactions take place at the electrodes, where an electric current is

produced. The ion species and its transport direction can differ. The ion can either be

negative or positive, which means that the ion carries either a negative or a positive

charge.

Figure 1.1: Schematic of an individual fuel cell [1].

The basic principles of a fuel cell are similar to the electrochemical batteries, which are

involved in many activities of dailylife. The main difference between the batteries and

the fuel cells is that, in the case of batteries, the chemical energy is stored in substances

located inside them. When this energy has been converted to electrical energy, the

battery must be thrown away ( primary batteries ) or recharged appropriately ( secondary

batteries ). In a fuel cell, on the other hand, since the chemical energy is provided by a

fuel and an oxidant stored outside the cell in which the chemical reaction takes place, the

electrical energy is produced for as long as the fuel and oxidant are supplied to the

electrodes. Figure 1.2 shows the comparison of direct energy conversion with fuel cells

to indirect conversion.

2

Chemical energy of the fuel(s)

Electrical energy conversion

Thermal and/or mechanical energy conversion

Figure 1.2: Direct energy conversion with fuel cells in comparison to indirect energy conversion.

Gaseous hydrogen has become the fuel of choice for most applications, because of

its high reactivity when suitable catalysts are used, its ability to be produced from

hydrocarbons and its high energy density when stored cryogenically for closed

environment applications. Similarly, the most common oxidant is gaseous oxygen,

which is readily and economically available from air, and easily stored in a closed

environment.

The electrolyte not only transports dissolved reactants to the electrode, but also

conducts ionic charge between the electrodes and thereby completes the cell electric

circuit. It also provides a pysical barrier to prevent the mixing of the fuel and oxidant gas

streams.

The porous electrodes in the fuel cells provide a surface site where gas/liquid

ionization or de-ionization reaction can take place. In order to increase the rates of

3reactions, the electrode material should be catalytic as well as conductive, porous rather

4

.2 Fuel Cell Plant Description

he fuel and oxygen from the air are combined

.3 Fuel Cell Stacking

r to produce the required voltage level. The

lar plate between

conditions, and an excellent electronic conductor. [2]

than solid. The catalytic function of electrodes is more important in lower temperature

fuel cells and less so in high temperature fuel cells, because ionization reaction rates are

directly proportional with temperature ( i.e. increase with temperature ). The porous

electrodes also provide a physical barrier that separates the bulk gas phase and the

electrolyte.

1

In the fuel cell, hydrogen produced from t

to produce dc power, water, and heat. In cases where CO and CH4 are reacted in the cell

to produce H2, CO2 is also a product. These reactions must be carried out at a suitable

temperature and pressure for fuel cell operation. A system must be built around the fuel

cells to supply air and clean fuel, convert the power to a more usable form

( i.e. AC power ), and remove the depleted reactants and heat that are produced by the

reactions in the cells. Figure 1.3 shows a simple rendition of a fuel cell power plant.

Beginning with fuel processing, a conventional fuel ( natural gas, other gaseous

hydrocarbons, methanol, coal, etc. ) is cleaned, then converted into a gas containing

hydrogen. Energy conversion occurs when dc electricity is generated by means of

individual fuel cells combined in stacks or bundles. A varying number of cells or stacks

can be matched to a particular power application. Finally, power conditioning converts

the electric power from DC into regulated DC or AC for consumer use [1].

1

Individual fuel cells are combined in orde

schematic of stacking of individual fuel cells is given in Figure 1.4. [2]

Anode – electrolyte – cathode sections are connected in series by a bipo

the cathode of the cell and the anode of the other cell. The bipolar plate must be

impervious to the fuel and oxidant gases, chemically stable under reducing and oxidizing

Fuel Processor

Power Section

Power Conditioner

Natural Gas

H Gas

5

igure 1.4 is a representation of a flat plate cell. Tubular solid oxide cells are stacked in

different way. There may be other arrangements for stacking as well, provided that the

Figure 1.3: Fuel Cell Power Plant Processes

F

a

interconnectors are impervious to the gases and are excellent electronic conductors, as

explained.

2-richDC Power

AC Power

ir

Exhaust Gas

A

Usable Heat

Steam

Figure 1.4: Stacking of individual fuel cells. [2]

.4 Characteristics of Fuel Cells

These advantages of fuel cells are grouped

.4.1 Efficiency

ical energy is converted directly into electrical energy. Since direct energy

1

Fuel cells offer advantages in many fields.

into categories and are briefly explained in this section.

1

Chem

conversion doesn’t require a preliminary conversion into heat, this conversion is not

subject to the limitations of Carnot cycle, and thermal efficiencies of as high as 90% are

theoretically possible. This direct energy conversion from chemical energy to electrical

6

7

.4.2 Flexibility in Power Plant Design

e o low voltage level of an individual cell, it is

.4.3 Manufacturing and Maintenance

a low as engines. The whole system of the fuel

.4.4 Noise

l s no moving parts. It runs quietly, does not vibrate, does not generate

energy does not require any mechanical conversion, such as boiler-to-turbine and

turbine-to-generator systems. The efficiency of a cell is not dependent upon the size of

the cell. A small cell operates with an efficiency equivalent to a larger one, consequently

can be just as efficient as large ones. This is very important in the case of the small local

power generating systems needed for combined heat and power systems.

1

In ord r to obtain a desired voltage, due t

necessary to connect a number of cells in series. The current delivered by an individual

cell is proportional to the geometrical area of the electrode. The electrode may be

increased in size, or alternatively, several cells may be connected in parallel to increase

the current. These cell groups may also be connected in series or parallel to yield high

currents at high voltages. The cells need not be localized in one place, thus providing

flexibility in weight distribution and space utilization. This characteristics is most

convenient from a design viewpoint.

1

The m nufacturing cost of fuel cells is as

cells can be manufactured by mass production methods. There are no moving parts in a

cell, hence sealing problems are minimum and no bearing problems exist. Because of

these, fuel cells require little or no maintenance. Corrosion, on the other hand, is a

serious problem, especially for high temperature cells.

1

A fue cell ha

gaseous pollutants [2]. This characteristics is very important in military and

communication applications.

8

.4.5 Heat

e inefficiencies in a fuel cell may manifest themselves as heat. With proper

.4.6 Low Emissions

y main fuel cell reaction, when hydrogen is the fuel, is water,

ffer can be listed as

el flexibility

ty

pability

General negative features of fuel cells and fuel cell plants can be listed as follows :

iliar technology to the power industry

1

The el ctrical

design of a cell, efficiency can be maximized and heat can be minimized [3].

1

The b -product of the

instead of carbon dioxide, nitrogen oxides, sulfur oxides, and particulate matter inherent

to fossil fuel combustion, which means a fuel cell can be essentially a “zero emission”

device. This is fuel cells’ main advantage when used in vehicles, as there is a

requirement to reduce vehicle emissions, and even eliminate them within cities.

However, it should be noted that, at present, emissions of CO2 are nearly always

involved in the production of the hydrogen needed as the fuel [4].

Some other characteristics that fuel cells and fuel cell plants o

follows :

• Fu

• Cleanliness

• Size flexibili

• Cogeneration ca

• Site flexibility

• Cost

• Unfam

9

1.5 Types of Fuel Cells and Their Fields of Applications

Fuel cells are usually classified, according to their operating temperatures, into low,

medium, and high temperature fuel cells. Table 1.1 gives an overview of the fuel cell

technologies presently under development [5].

Table 1.1: Classification of fuel cells. [5]

Fuel Cell Type

Operating

Temperature

[oC]

Fuel Oxidation

Media

Typical Unit

Sizes

[kWe]

Alkaline 70 - 100 H2 Oxygen << 100

Protone Exchange

Membrane 50 - 100

H2 and reformed

H2

Oxygen

from air 0,1 - 500

Phosphoric Acid 160 - 210 H2 reformed from

natural gas

Oxygen

from air

5 - 200

(plants up to

5,000)

Molten Carbonate 650

H2 and CO from

internal reforming

of natural/coal gas

Oxygen

from air

800 - 2,000

(plants up to

100,000)

Solid Oxide 800 - 1000

H2 and CO from

internal reforming

of natural/coal gas

Oxygen

from air 2.5 - 100,000

10

The most probable application fields for the different types of fuel cells are presented in

Table 1.2.

Table 1.2: Application fields of fuel cells.

Fuel Cell Type Fields of Application

Alkaline Space applications and special military applications

Protone Exchange

Membrane

Stationary applications for direct hydrogen use

Stationary applications for power and heat production

Mobile applications for buses, service vehicles

Mobile applications for railroad systems

Mobile applications for passenger cars

Phosphoric Acid Stationary applications for power and heat production


Molten Carbonate Stationary applications for combined power and vapor production

Stationary applications for utility use

Solid Oxide

Stationary applications for power and heat production

Stationary applications for utility use


CHAPTER 2

THERMODYNAMICS OF FUEL CELLS

2.1 Some Fundamental Relations

2.1.1 TdS Equations and Maxwell Relations

Consider a simple compressible system undergoing an internally reversible process. An

energy balance for this simple compressible system, in the absence of overall system

motion and gravity effect, can be written in differential form as follows;

..int..int revrev WdUQ δδ += ( 2.1 )

The only mode of energy transfer by work that can occur as a simple compressible

system undergoes quasiequilibrium processes is associated with volume change and is

given by ∫ [6]. Therefore, the work is given by pdV

pdVW rev =..intδ ( 2.2 )

The equation for entropy change on a differential basis is given by

..int revTQdS δ

= ( 2.3 )

11

By rearrangement,

TdSQ rev =..intδ ( 2.4 )

Substituting Eqs.2.2 and 2.4 into Eq.2.1 and rearranging the terms gives the first TdS

equation;

pdVTdSdU −= ( 2.5 )

Enthalpy is, by definition,

pVUH += ( 2.6 )

On a differential basis,

VdppdVdUpVddUdH ++=+= )( ( 2.7 )

Rearranging the terms results,

VdpdHpdVdU −=+ ( 2.8 )

Substituting Eq.2.8 into Eq.2.5 and rearranging the terms gives the second TdS equation;

VdpTdSdH += ( 2.9 )

The TdS equations on a unit mass basis can be written as

pdvTdsdu −= (2.10)

vdpTdsdh += (2.11)

or on a per mole basis as

vpdsTdud −= (2.12)

pvdsTdhd += (2.13)

From these two fundamental relations, two additional equations may be formed by

defining two other properties of matter.

The Helmholtz function ψ is defined by the equation

Tsu −=Ψ (2.14)

12

Forming the differential dψ results,

sdTTdsduTsddud −−=−=Ψ )( (2.15)

Substituting Eq.2.10 into Eq.2.15 gives

sdTpdVd −−=Ψ (2.16)

The Gibbs function is defined by the equation

Tshg −= (2.17)

Forming the differential dg results,

sdTTdsdhTsddhdg −−=−= )( (2.18)

Substituting Eq. 2.1 into Eq. 2.8 gives

sdTvdpdg −= (2.19)

From the comparison of Eq.2.16 and Eq.2.19, one can conclude that Gibbs function

carries out reactions at constant pressure and temperature, while Helmholtz function

does at constant volume and temperature. Since it is more practical to carry out reactions

at constant pressure and temperature, Gibbs function is more useful and is preferred in

calculations.

As a result, the summary of these four important relationships among properties of

simple compressible systems are collected and presented below:

pdvTdsdu −= (2.10)

vdpTdsdh += (2.11)

sdTpdVd −−=Ψ (2.16)

sdTvdpdg −= (2.19)

These equations are referred to as TdS ( or Gibbsian ) equations. Note that the variables

on the right-hand sides of these equations include only T, s, p, and v.

13

Consider three thermodynamic variables represented by x, y, and z. Their functional

relationship may be expressed in the form x = x ( y, z ). The total differential of the

dependent variable x is given by the equation

dzzxdy

yxdx

yz

⎟⎠⎞

⎜⎝⎛∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

= (2.20a)

If in Eq.2.20a we denote the coefficient of dy by M and the coefficient of dz by N,

Eq.2.20a becomes

NdzMdydx += (2.20b)

Partial differentiation of M and N with resprect to z and y, respectively, leads to

zyx

zM

y ∂∂∂

=⎟⎠⎞

∂∂ 2

(2.21a)

and

yzx

yN

z ∂∂∂

=⎟⎟⎠

⎞∂∂ 2

(2.21b)

If these partial derivatives exist, it is known from the calculus that the order of

differentiation is immaterial, so that

zy yN

zM

⎟⎟⎠

⎞∂∂

=⎟⎠⎞

∂∂ (2.21c)

When Eq.2.21c is satisfied for any function x, then dx is an exact differential. Eq.2.21c

is known as the test for exactness. [7]

Since only properties are involved, each TdS equation is an exact differential exhibiting

the general form of Eq.2.20a. Underlying these exact differentials are functions of the

form u ( s, v ), h ( s, p ), ψ ( v, T ), and g ( T, p ), respectively.

The differential of the function u ( s, v ) is

dvvuds

sudu

sv⎟⎠⎞

⎜⎝⎛∂∂

+⎟⎠⎞

⎜⎝⎛∂∂

= (2.22)

14

Comparing Eq.2.22 to Eq.2.10 results,

vsuT ⎟⎠⎞

⎜⎝⎛∂∂

= (2.23a)

svup ⎟⎠⎞

⎜⎝⎛∂∂

=− (2.23b)

The differential of the function h ( s, p ) is

dpphds

shdh

sp⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+⎟⎠⎞

⎜⎝⎛∂∂

= (2.24)


pshT ⎟⎠⎞

⎜⎝⎛∂∂

= (2.25a)

sphv ⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

= (2.25b)

The differential of the function ψ ( v, T ) is

dTT

dvv

dvT⎟⎠⎞

⎜⎝⎛∂Ψ

+⎟⎠⎞

⎜⎝⎛∂Ψ∂

=Ψα (2.26)


Tvp ⎟

⎠⎞

⎜⎝⎛∂Ψ∂

=− (2.27a)

vTs ⎟

⎠⎞

⎜⎝⎛∂Ψ∂

=− (2.27b)

The differential of the function g ( T, p ) is

dTTgdp

pgdg

pT

⎟⎠⎞

⎜⎝⎛∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

= (2.28)

15


Tpgv ⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

= (2.29a)

pTgs ⎟⎠⎞

⎜⎝⎛∂∂

=− (2.29b)

Since each of the four differentials is exact and similar to Eq.2.20a, referring to

Eq.2.21c, the following relations can be written:

vs sp

vT

⎟⎠⎞

⎜⎝⎛∂∂

−=⎟⎠⎞

⎜⎝⎛∂∂

(2.30)

ps sv

pT

⎟⎠⎞

⎜⎝⎛∂∂

=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

(2.31)

Tv vs

Tp

⎟⎠⎞

⎜⎝⎛∂∂

=⎟⎠⎞

⎜⎝⎛∂∂

(2.32)

Tp ps

Tv

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−=⎟⎠⎞

⎜⎝⎛∂∂

(2.33)

This set of equations is referred to as the Maxwell relations.

2.1.2 Partial Molal Properties

In general, the change in any extensive thermodynamic property X of a multicomponent

system can be expressed as a function of two independent intensive properties and size

of the system. Selecting temperature and pressure as the independent properties and the

number of moles n as the measure of size, this change in any extensive thermodynamic

property X can be expressed as follows:

inpTi inTnp

dnnXdp

pXdT

TXdX

j,,,,∑ ⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+⎟⎠⎞

⎜⎝⎛∂∂

= (2.34)

16

where the subscript nj denotes that all n’s except ni are held fixed during differentiation.

The last term on the right-hand side of the Eq.2.34 is defined as the partial molal

property iX of the ith component in a mixture. Therefore, the partial molal property

iX is, by definition

jnpTii n

XX,,

⎟⎟⎠

⎞∂∂

= (2.35)

The extensive thermodynamic property X, can be expressed in terms of the partial molal

property iX as

∑=j

ii XnX1

(2.36)

Eq.2.36 can be referred in order to evaluate the change in volume on mixing of pure

components which are at the same temperature and pressure. Selecting V as the

extensive property X in Eq.2.36 the total volume of the pure components before mixing

is

∑=

=j

iiicomp vnV

1,0. (2.37)

where iv ,0 is the molar specific volume of pure component i. The volume of the mixture,

using Eq.2.36, is

∑=

=j

iiimix vnV

1. (2.38)

where iv is the partial molal volume of component i in the mixture. Hence, the volume

change on mixing is given by

∑∑==

−=−=∆j

iii

j

iiicompmixmixing vnvnVVV

1,0

1.. (2.39a)

or

17

( )∑=

−=∆j

iiiimixing vvnV

1,0 (2.39b)

Selecting U, H, and S as the extensive properties, the similar results can be obtained as

follows:

( )∑=

−=∆j

iiiimixing uunU

1,0 (2.40a)

( )∑=

−=∆j

iiiimixing hhnH

1,0 (2.40b)

( )∑=

−=∆j

iiiimixing ssnS

1,0 (2.40c)

In Eqs.2.40a – c, iu ,0 , ih ,0 , and is ,0 denote molar internal energy, enthalpy, and entropy

of pure component i; iu , ih , and is denote respective partial molal properties.

2.1.3 Chemical Potential

Of the partial molal properties, the partial molal Gibbs function is particularly useful in

describing the behaviour of mixtures and solutions. This quantity plays a central role in

the criteria of both chemical and phase equilibrium. Because of its importance in study

of multicomponent systems, the partial molal Gibbs function of component i is given a

special name and symbol. It is called the chemical potential of component i and

symbolized by µi. [6]

jnpTiii n

GG,,

⎟⎟⎠

⎞∂∂

==µ (2.41)

Gibbs function can be expressed in terms of chemical potential as

∑=j

iinG1

µ (2.42)

18

The differential of G ( T, p, n1, n2, ... , nj ) can be formed as

∑ ⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+⎟⎠⎞

⎜⎝⎛∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

=i

inpTinpnT

dnnGdT

TGdp

pGdG

j,,,,

(2.43)

Substituting Eqs.2.29a – b into Eq.2.43 yields,

∑=

+−=j

iii dnSdTVdpdG

1µ (2.44)

2.2 Thermodynamics of Chemical Reactions

2.2.1 Free Energy Change of Chemical Reactions

Consider the chemical reaction below;

dDcCbBaA +→+ (2.45)

The change in Gibbs function of reaction, or Gibbs free energy of the reaction, under

constant temperature and pressure, is given by the equation

BADC badcG µµµµ −−+=∆ (2.46)

where µ is the chemical potential of the species.

The maximum net work obtainable from a chemical reaction can be calculated by the

free energy change of the chemical reaction. Referring to Eq.2.17, the free energy

change of a chemical reaction is given by,

STHG ∆−∆=∆ (2.47)

2.2.2 Standard Free Energy Change of a Chemical Reaction

The chemical potential of any substance may be expressed by an equation of the form

aRTo ln+= µµ (2.48)

19

where a is the activity of the substance and µ has the value µ° when a is unity. The

standard free energy change ∆G° of the reaction 2.43 is given by

oooo BADCo badcG µµµµ −−+=∆ (2.49)

where µC° indicates the standard chemical potential of product C, and so on. Substituting

Eqs.2.48 and 2.49 into Eq.2.46 yields

bB

aA

dD

cCo

aaaa

RTGG ln+∆=∆ (2.50)

Hence, the standard free energy change of a chemical reaction is

bB

aA

dD

cCo

aaaaRTGG ln−∆=∆ (2.51)

Assuming a process at constant temperature and pressure at equilibrium, since the free

energy change for this process is zero, Eq.2.51 becomes

KRTaaaa

RTG beqB

aeqA

deqD

ceqCo lnln

,,

,, −=−=∆ (2.52)

where the suffixes eq in the activity terms indicate the values of the activities at

equilibrium, and K is the equilibrium constant for the reaction.

The importance of the knowledge of ∆G° is that it allows ∆G to be calculated for any

composition of a reaction mixture. Knowledge of ∆G indicates whether a reaction will

occur or not. If ∆G is positive, a reaction cannot occur for the assumed composition of

reactants and products. If ∆G is negative, a reaction can occur. [8]

2.2.3 Relation Between Free Energy Change in a Cell Reaction and Cell Potential

The enthalpy change of any reaction, assuming constant temperature and pressure, can

be showed as follows :

VPWQVPEH ∆+−=∆+∆=∆ (2.53)

20

If the reaction is carried out in a heat engine, then the only work done by the system

would be the expansion work,

VPW ∆= (2.54)

Hence Eq.2.51 becomes;

QH =∆ (2.55)

If the same reaction, which is under consideration is carried out electrochemically, the

only work done by the system will not be the expansion work of the gases produced, but

will also be the electrical work due to the charges being transported around the circuit

between the electrodes. The maximum electrical work that can be done by the overall

reaction carried out in a cell, where Vrev,c and Vrev,a are the reversible potentials at the

cathode and anode respectively, is given by

( )arevcreve VVneW ,,max, −= (2.56)

In the cell, n electrons are involved and the cell is assumed to be reversible

( i.e., overpotential losses are assumed to be zero ). Multiplying Eq.2.56 by the

Avogadro number, N, in order to have molar quantities gives;

reve VnFW ∆=max, (2.57)

where F is the Faraday number, and revV∆ is the difference between reversible electrode

potentials.

The only work forms assumed are the expansion work and electrical work.

VPWW e ∆+= max, (2.58)

In addition to these, assuming the process is reversible

STQ ∆= (2.59)

Substituting Eqs.2.57 – 2.59 into Eq.2.53, the enthalpy change will be

revVnFSTH ∆−∆=∆ (2.60)

21

Eq.2.60 can be rearranged as follows,

revVnFSTH ∆−=∆−∆ (2.61)

where

STHG ∆−∆=∆ (2.47)

and

revVE ∆= (2.62)

Substituting Eqs.2.47 and 2.62 into Eq.2.61 gives

nFEG −=∆ (2.63)

E, which is defined as the difference in potentials between the electrodes is called as the

electromotive force of the cell ( i.e, the reversible potential of the cell, Erev ). If both the

reactants and the products are in their standard states, Eq.2.63 can be written as,

oo nFEG −=∆ (2.64)

where E° is the standard electromotive force, or – as most commonly referred to – is the

standard reversible potential of the cell.

2.3 Nernst Equation

Let us consider the following reaction,

mMlLkK →+ (2.65)

where k moles of K react with l moles of L to produce m moles of M. Each of the

reactants and the products have an associated activity; aK , and aL being the activity of

the reactants, aM being the activity of the product. For ideal gases, activity term can be

written as

(2.66) 0p

pa =

22

p is the partial pressure of the gas, and p0 is the pressure of the cell. Eq.2.50 can be

rearranged for the the reaction given in Eq.2.65, as follows.

⎟⎟⎠

⎞⎜⎜⎝

⎛+∆=∆ l

LkK

mMo

aaaRTGG ln (2.67)

In the Eq.2.67, G∆ and oG∆ show the change in molar Gibbs free energy of formation,

and the change in standard molar Gibbs free energy of formation.

From Eq.2.63, the following relation can be written,

nFGE ∆

−= (2.68)

Substituting Eq.2.68 into Eq.2.67 gives the effect on voltage as follows,

⎟⎟⎠

⎞⎜⎜⎝

⎛−

∆−= l

LkK

mM

o

aaa

nFRT

nFGE ln (2.69)

Substituting Eq. 2.64 into Eq 2.69 yields,

⎟⎟⎠

⎞⎜⎜⎝

⎛−= l

LkK

mMo

o aaa

nFRTEE ln (2.70a)

where E° is the standard electromotive force, and Eo is defined to indicate the reversible

electric voltage. Eq.2.70a can be rewritten by substituting Eq.2.52 and Eq.2.66, as

follows.

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛

−= l

o

L

k

o

K

m

o

M

o

pp

pp

pp

nFRTK

nFRTE lnln (2.70b)

Eq. 2.70a and 2.70b give the electromotive force in terms of product or/and reactant

activity, and is called Nernst equation. The electromotive force calculated using this

equation is known as the Nernst voltage, and is the reversible cell voltage that would

exist at a given temperature and pressure.

23

2.4 Exergy Concept

2.4.1 Exergy Balance

Exergy is defined as the maximum amount of work obtainable a substance can yield

when it is brought reversibly to equilibrium with the environment [9]. The exergy

analysis of a system is based on the second law of thermodynamics and the concept of

entropy production. In order to describe the exergy concept, a fuel cell can be modeled

as a control volume of a thermodynamic system with a single inlet and outlet, as shown

in Figure 2.1.

To

24

Figure 2.1 : An open thermodynamic system with single inlet and outlet.

inlet outlet

oQ& W&

Environment jQ&

kiTP iooo

,,1,, ,

K=

µ lj ,,1K= Tj

The goal in power producing systems is to maximize net work and efficiency. A power

plant operates according to the first and second laws of thermodynamics [10]. To

calculate the maximum work that can be produced, let us consider the system in Figure

2.1. The streams in and out of the system consist of n species with molar flow

rates , where i = 1… k. The heat transfer interactions … and properties at

the inlet and outlet are assumed to be fixed. [11] The first law for the system in Figure

2.1 can be written as;

outiini nn ,, , && 1Q& lQ&

( ) ( )∑∑∑===

−−−+−+=k

ioutioutiit

k

iiniiniit

l

jj nhhnhhWQQ

dtdE

1,,0,

1,,0,

10 &&&&& (2.71)

The second law for the same system can be written as;

( ) ( ) gen

k

ioutioutii

k

iiniinii

l

j j

j SnssnssTQ

TQ

dtdS &&&

&&+−−−+⎟

⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛= ∑∑∑

=== 1,,0

1,,0

10

0 (2.72)

In the equations above, ht,i is the total specific enthalpy and is

iit gzVhh ⎟

⎠⎞

⎜⎝⎛ ++= 2

, 21 (2.73)

½V2 is the kinetic energy, gz is the potential energy of the mass flow, and these kinetic

and potential energies may be neglected so that ht,i = hi. E is the total energy of the

system, and S is the entropy of the system. In order to make time derivatives zero,

system is assumed to be steady state, steady flow. By eliminating between Eqs.2.71

and 2.72, the exergy balance can be obtained. Therefore, the exergy balance is given by

the following equation.

0Q&

( )[ ]

( )[ ] gen

k

ioutiioutii

k

iiniiinii

l

jj

j

STnsThW

nsThQTT

&&&

&&

01

,,00

1,,00

1

01

+⋅−−+=

⋅−−+⎟⎟⎠

⎞⎜⎜⎝

⎛−

∑

∑∑

=

==

µ

µ

(2.74)

25

W& is the actual work of the system. It can be noted that entropy generation reduces the

available work, as is expected. The exergy balance for an open system in Eq.2.74 shows

that the exergies in heat flows ( the first term on the left hand side of the equation ) and

mass flows ( the second term on the left hand side of the equation ) supplied to the

system are equal to the work produced ( the first term on the right hand side of the

equation ), exergy in the outlet mass flow ( the second term on the right hand side of the

equation ), and exergy destroyed through irreversible processes ( the third term on the

right hand side of the equation ) [11]. For the steady state, steady flow system, energy

input is equal to the energy output. Due to irreversible processes, outlet exergy is always

less than the inlet exergy. Exergy destruction is a result of chemical and physical

processes that take place in the system.

Irreversibility, for a system can be described as the difference between the reversible

work ( maximum work that can be obtained ) and the actual work. Hence, from the

definition

genactrev STWWI &&&&0=−= (2.75)

The exergy balance ( Eq.2.74 ) can be used to calculate the irreversibility.

Exergy analysis requires that the environment is defined. For a general case,

environment can be assumed to be at standard temperature and pressure conditions

( i.e., T = 298 K, P = 1 atm ), but this assumption is not always the case. Environment

definition can differ from system to system.

2.4.2 Chemical Exergy

Chemical exergy is equal to the maximum amount of work obtainable when the

substance under consideration is brought from the environmental state to the dead state

by processes involving heat transfer and exchange of substances only with the

environment. [9]

Chemical exergy of a mixture is given by the following equation.

26

∑∑ +=i

iioi

oiimix xxRTxEx ln~ε (2.76)

oiε

~ is the standard chemical exergy of substance i. [9] The exergy of the mixture is

always less than the sum of the exergies of its components at the temperature and

pressure of the mixture, since the second term on the right hand side is always negative.

2.4.3 Physical Exergy

Physical exergy is equal to the maximum amount of work obtainable when the stream of

substance is brought from its initial state to the environmental state defined by Po and To

by physical processes involving only thermal interaction with the environment. [9]

Defining the environment state at Po, To, assuming the kinetic and potential energies are

negligible, the physical exergy of a substance at state P1, T1 is calculated by the

following equation.

( ) ( ooooph sThsThEx −−−= 111, ) (2.77)

2.5 Efficiency of Fuel Cells

2.5.1 Thermodynamic ( First and Second Law ) Efficiencies

In order to define the efficiency of fuel cells, let us consider a simple H2/O2 fuel cell,

operating at T = 25°C. and P = 1 atm. as shown in Figure 2.2. The inlet and outlet

conditions are assumed to be the same for simplicity.

The energy balance for the fuel cell is,

)()(21)(

222ThWThThQ OHOH +−=++ && (2.78a)

27

28

Figure 2.2 : Simple H2/O2 fuel cell system, T = 25°C.

The entropy balance for the fuel cell is,

OHOH sTsTsTQ

222)(

21)( =++

& (2.78b)

The energy and entropy balance can be rewritten respectively, as follows,

WHQ && −∆= (2.79a)

STQ ∆=& (2.79b)

Combining energy and entropy balances yields,

WHST &−∆=∆ (2.80)

( )TH 2

( )TO2½

( )TOH 2

W&−

Q&

T

Hence, work output of the system is,

STHW ∆−∆=& (2.81)

The first law efficiency is defined as the ratio of the work output of the system to energy

input to the system. The work output of the system is equal to the free energy change

( i.e., Gibbs function ). The energy input to the system, is the chemical energy of the

fuel. Therefore the energy input is,

HQ ∆=& (2.82)

Hence, the first law efficiency is given by,

HST

HSTH

QW

I ∆∆

−=∆

∆−∆== 1

&

&η (2.83)

The second law efficiency is defined as the ratio of the work output of the system to the

maximum work output ( i.e., reversible work ) of the system. To determine the second

law efficiency, let us assume that the fuel cell is adiabatic, and no work interactions

occur inside the cell. Hence, the energy input to the system is unchanged at the outlet of

the fuel cell. If we can consume all of this energy and change it to work, then we can

determine the reversible work output of the system. A model for this study is shown in

Figure 2.3.

The reversible heat exchanger, added to the exit of the fuel cell, operates between the

fuel cell outlet temperature and the environment temperature To ( which usually

is 25°C ). The heat consumed is sent to a reversible heat engine, operating between the

heat exchanger and the environment. The work obtained by the heat engine is the

reversible work output of the system.

The subscripts “R” and “P” are used in order to indicate the reactants, and the products

respectively.

29

30

Figure 2.3 : Schematic of the system used to calculate the second law efficiency of the simple fuel cell.

Now, let us apply energy balance to the control volume 1.

)()( 32 THQTH PP += & (2.84)

Applying the entropy balance to the control volume 2 gives,

)()( 32 TSTQ

TS Po

oP +=

& (2.85)

Fuel Cell

Heat Exchanger

Heat Eng.

1 2 3

To

To

Q&

oQ&

revW&

Control Volume 1

adiabatic

Control Volume 2

The energy balance for the heat engine can be written as,

orev QWQ &&& += (2.86)

Therefore, substituting Eqs.2.84 and 2.85 into Eq.2.86, the reversible work output can be

obtained.

( ) ( ))()()()( 3232 TSTSTTHTHW PPoPPrev −−−=& (2.87a)

STHW orev ∆−∆=& (2.87b)

The work output of the system was found by Eq.2.81 as

STHW ∆−∆=& (2.81)

Therefore, the second law efficiency for the fuel cell can be written as follows.

STHSTH

WW

orevII ∆−∆

∆−∆==

&

&η (2.88)

One of the advantages of the fuel cells is their high efficiency, as mentioned before.

Using Eq.2.83 and 2.88, the change of first and second law efficiencies with temperature

is graphed and is given in Figure 2.4. Referencing Figure 2.4, we can conclude that fuel

cell has higher efficiency at lower temperatures. As the temperature increases, the

efficiency decreases. This is the main difference between fuel cell efficiency and the

Carnot efficiency. The Carnot efficiency, by which thermal engines are compared in

their efficiency, for a thermal engine operating at temperature T, is given by,

TTo

C −= 1η (2.89)

To is the environment temperature where thermal engine is working. As the working

temperature of thermal engine increases, since the second term of the Carnot efficiency

will approach zero, the Carnot efficiency increases as well. The comparison of fuel cell

efficiencies and the Carnot efficiency with temperature is shown in Figure 2.4.

Fuel cell efficiency decreases with increasing temperature, while the Carnot efficiency

increases with increasing temperature. Investigating this graph only, gives the low

31

temperature fuel cells have high efficiencies than the high temperature ones. But,

although the graph suggests that lower temperature fuel cells are better, the voltage

losses are usually less at high temperatures.* So, in general, fuel cell voltages are usually

higher at high temperatures. On the other hand, the waste heat from the high temperature

fuel cells is more useful than the waste heat from the low temperature fuel cells.

Therefore, only this graph itself cannot be referenced to make a decision on fuel cells

working at different conditions.

0,0

0,2

0,4

0,6

0,8

1,0

0 200 400 600 800 1000 1200

Temperature [ °C ]

Effic

ienc

y ( %

)

Carnot Ef f .

F.C.Ist Law Ef f ic iency

F.C. IInd Law Ef f iciency

Figure 2.4: Comparison between fuel cell first law and second law efficiency changes with temperature and Carnot efficiency change with temperature.

32* Voltage losses and causes of voltage drops are discussed in Chapter 3.

2.5.2 Electrochemical Efficiencies

The efficiency term for fuel cells, given by the Eq. 2.81, can be written in terms of the

reversible electrode potentials, using Eq. 2.63, the ideal efficiency is given as follows,

HnFV rev

i ∆−

=η (2.90)

Vrev is the reversible potential of the cell, which is the ideal case. When the fuel cell is

under load, the actual potential of the cell, Vact, will fall below the reversible potential,

due to the irreversibilities. Hence, the actual efficiency will be,

HnFV act

ac ∆−

=η (2.91)

These irreversibilities are unwanted effects in the cell, since they decrease the reversible

potential. As a result, the reversible work of the system will decrease. This difference

between the reversible work and the actual work is the heat rejected from the system.,

and is larger than the reversible heat transfer T∆S.

The ratio of the actual potential of the cell to the reversible potential of the cell is

defined as the voltage efficiency, ηv. Hence,

rev

actv V

V=η (2.92)

When the fuel reacts electrochemically in the cell, some fuels may react directly to give

heat release in the cell or may react to products other than those required, hence do not

take place in the electrochemical reaction. Considering the total number of moles of

fuels, being reacted electrochemically, the faradaic efficiency, ηF can be expressed as

follows.

fF NnF

i&

=η (2.93)

fN& is the total number of moles of fuel reacted electrochemically per second. ηF is the

fraction of reaction which is occurring electrochemically to give current.

33

34

CHAPTER 3

KINETIC EFFECTS

3.1 Introduction

In order for a chemical reaction to be considered as a source of energy in a fuel cell,

there are two criteria that must be satisfied. First, at least one of the reactants must be

ionizable at the operating conditions. The formation of ions results in the establishment

of an oxidation – reduction potential which, when traversed by the ions, ultimately

provides the energy that will be used at the terminals. Second, while the ionizable

reaction system is the source of energetic electron flow that does work in the external

circuit, a high flow rate is required for practical purposes. [3] As a result, rapid rates for

the electron supplying and consuming reactions are a second criterion.

This second criterion is the subject of chemical kinetics and is dynamic in nature. On the

other hand, the first criterion is associated with a static or equilibrium system and is

often the object of thermodynamics inquiry. The equal rates of the forward and

backward electrode reactions that occur at static conditions establish the equilibrium.

Therefore, this equilibrium may be considered dynamic and it may be helpful to

consider it a problem in kinetics.

The open circuit voltage of a fuel cell is given by the following formula,

nFGE ∆

−= (2.66)

This is the theoretical value of the open circuit voltage, which is also referred to as the

reversible open circuit voltage.

Whenever a load is applied to a cell in which the electrodes are reversible, it causes its

electrodes to shift its potential, i.e. polarize in opposite direction. As a result of this

polarization, the cathodes become less cathodic and the anodes become less anodic,

resulting a decrease in the available cell voltage. Therefore, the operating voltage is less

than the reversible voltage and this is because of the losses or irreversibilities.

Figure 3.1 shows the performance of a simple fuel cell operating at about 40°C, at

normal air pressure [4].

0

0,2

0,4

0,6

0,8

1

1,2

1,4

0 200 400 600 800 1000

Current Density ( mA/cm² )

Cel

l Vol

tage

( Vo

lts )

Reversible Cell Voltage ( no loss )

Actual Cell Voltage

Figure 3.1: Voltage change with current density for a simple fuel cell operating at about 40°C, and at standard pressure. [4]

35

By examining Figure 3.1, some key points may be listed as follows:

• The open circuit voltage is less the theoretical open circuit voltage.

• There is a rapid initial fall in voltage.

• After this rapid fall, voltage loss is less slowly, and more linearly.

• At some higher current density, voltage falls rapidly.

Another graph showing the voltage change with current density for a fuel cell, which is a

solid oxide fuel cell this time, is shown in Figure 3.2, in order to show the effect of high

temperature on circuit voltage change. From this graph, the following key points may be

listed:

0

0,2

0,4

0,6

0,8

1

1,2

0 200 400 600 800 1000

Current Density ( m A/cm ² )

Cur

rent

Vol

tage

( Vo

lts )

Reversible Cell Voltage ( no loss )

Actual Cell Voltage

Figure 3.2: Voltage change with current density for a solid oxide fuel cell operating at about 800°C.

36

• The open circuit voltage is equal to the theoretical open circuit voltage, or there is

only a very little difference

• Initial fall of voltage is very small, the graph is much more linear

• At some higher current density, voltage falls rapidly.

Comparison between Figures 3.1 and 3.2, it can be shown that the high temperature fuel

cells have lower reversible cell voltage, while they can have high operating voltages

since the voltage drop is smaller.

Examining Figures 3.1 and 3.2, the difference between the reversible cell voltage and

the actual voltage can be noticed. This difference, which grows as the current density is

increased is called the voltage drop. Voltage drop is the result of the irreversibilities in

the cell. These irreversibilities are the main subject of this chapter. The effects which

cause the actual voltage fall below the reversible voltage will be considered.

3.2 Fuel Cell Irreversibilities

If the cell is reversible, there will not be any voltage drop and the electric voltage will be

determined from the Nernst equation. The electric voltage, due to the irreversibilities

occuring in the cell decreases from its ideal value ( i.e., reversible value ). This relation

can be shown as follows.

η−= oEE (3.1)

η is the sum of the irreversibilities in the cell.

The causes of voltage drop in the fuel cell can be the results of three major

irreversibilities. These irreversibilities are explained in details in this section.

37

3.2.1 Activation Polarization

When the forward and backward reactions occur at equal rates and the currents

associated with these reactions are equal and are in opposite direction, electrode

processes are said to be in a state of reversible equilibrium, in which no net reaction

takes place. This state of reversible equilibrium is satisfied unless a current flows in an

external circuit connecting the two electrodes of the cell. Due to this reversible state, for

a current to flow, a net reaction must occur at each electrode and the forward and

backward reaction rates at each electrode cannot be equal.

In many chemical reactions the reacting species have to overcome an energy barrier in

order to react. This energy barrier is called the “activation energy” and results in

activation polarization.

Consider the elementary electrode reaction below;

PneR ↔+ (3.2)

where R shows the reactants and P shows the products, both of which include the

molecules and ions as well. For this reaction, the forward ( or the cathodic ) and the

backward ( or the anodic ) reaction rates can be written as follows:

Rff aAkV ⋅⋅= (3.3a)

Pbb aAkV ⋅⋅= (3.3b)

where A is the actual electrode surface area, kf and kb are the forwanrd anc backward

reaction rate constants per unit area, respectively, and aR and aP are the activities of the

reactants and products, respectively.

The relation between the electron flow ( or current ) and the reaction rate is given as,

nFVi = (3.4)

where F is the Faraday’s constant. The net current in the forward direction ( i.e. cathodic

current ) will then be the resultant of the forward and backward currents,

( ) bfbfc iiVVnFi −=−= (3.5a)

38

and the net current in the backward direction ( i.e. anodic current ) will be,

( ) fbfba iiVVnFi −=−= (3.5b)

Substituting Eqs.3.3a and 3.3b into Eq.3.5a, the net cathodic current becomes

( ) ( )[ ]PbRfc akaknFAi ⋅−⋅= (3.6)

Figures 3.1 and 3.2 show that the electrode potential decreases as the net current

increases. From this given fact, it is obvious that either the reactant and product

activities are not constant, or that potential and rate constants are interdependent.

Assuming that the activities of the reactants and the products can be held constant, the

relation of change in potential to current must depend on its relation to the reaction rate

constant. Since the reaction rates are functions of the activation energy, activation

energy may then be affected by potential.

The relation between reaction rate and activation energy is exponential and it is given as

follows:

⎟⎟⎠

⎞⎜⎜⎝

⎛ +°∆−⎟⎠⎞

⎜⎝⎛=

RTnFEG

ah

kTAV fRf

αexp (3.7)

where ∆Gf° is the standard free energy of activation for the forward reaction, k is

Boltzmann constant, h is Planck’s constant, and α is a proportionality factor which is

usually referred to as the transfer coefficient. Eq.3.7 can be rewritten in terms of reaction

rate constants for both forward and backward reactions as follows:

⎟⎟⎠

⎞⎜⎜⎝

⎛ +°∆−=

RTnFEG

hkTk f

f

αexp (3.8a)

( )⎟⎠⎞

⎜⎝⎛ −−°∆−

=RT

nFEGh

kTk bb

α1exp (3.8b)

where ∆Gb° is the standard free energy of activation for the backward reaction, and may

also be written as,

°∆+°∆=°∆ fb GGG (3.9)

39

From Eqs.3.6 and 3.8a and b, the rate of the forward reaction, expressed as current

density ic may be written as

( )⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛ −−°∆−°∆−−⎟⎟⎠

⎞⎜⎜⎝

⎛ +°∆−=

RTnFEGG

aRT

nFEGa

hkTnFAi f

Pf

Rc

αα 1expexp (3.10)

Substituting Eq.3.7 into Eq. 3.10 gives

( )⎭⎬⎫

⎩⎨⎧

⎟⎠⎞

⎜⎝⎛ −−

−⎟⎠⎞

⎜⎝⎛=

RTnFEV

RTnFEVnFi rfc

αα 1expexp (3.11)

Referring from Eq.3.1, polarization, or overpotential, can be written as,

0EE −=η (3.12)

where E0 is the reversible potential. Modifying Eq.3.11 in terms of polarization and the

reversible potential yields,

( ) ( )⎟⎠⎞

⎜⎝⎛ −−

⎟⎠

⎞⎜⎝

⎛ −−−⎟

⎠⎞

⎜⎝⎛⎟

⎠

⎞⎜⎝

⎛=RT

nFRT

nFEnFV

RTnF

RTnFE

nFVi bfcηααηαα 1exp

1expexpexp 00 (3.13)

At open circuit or reversible potential, the net cathodic and anodic currents are zero, i.e.,

Eq.3.13 equals zero. The cathodic current if and anodic current ib are equal. This current,

which flows with equal intensity anodically and cathodically, at Eo is specifically

identified as the exchange current, io [3]. Therefore, the net cathodic current can be

written as

( )⎥⎦⎤

⎢⎣⎡ −−

−=RT

nFRTnFii oc

ηαηα 1expexp (3.14a)

The transfer coefficient is considered to be the fraction of the change in polarization that

leads to a change in the reaction rate constant, and its value is usually 0.5 for the fuel

cell applications [12]. Therefore, Eq.3.14a may be written as,

⎥⎦⎤

⎢⎣⎡ −

−=RTnF

RTnFii oc 2

exp2

exp ηη (3.14b)

40

From calculus the following relation can be written.

( ) ( ) ( )bbb sinh2expexp

=−− (3.15)

where b is any variable. Using this general knowledge, Eq.3.14b can be rearranged in

terms of sinh function as follows,

⎟⎠⎞

⎜⎝⎛=

RTnFii oc 2

sinh2 η (3.16)

In the discussion up to this point, we assumed that the reaction occurs in a single step.

But, in reality, this is not the case all the time, the reaction may go through several

intermediate steps, each with an associated energy barrier. The step with the highest

energy barrier within these intermediate steps is usually assumed to be the rate

determining step. Hence the other steps will be in an equilibrium state. The rate

determining step, however, may involve fewer electrons than the overall reaction. For

this reason, for the Eqs.3.4 – 3.16, the number of electrons term “n” must be replaced

with the number of electrons transferred in the rate determining step “ne” and ne must be

lower than or equal to n [13].

The cathodic current and the anodic current are the same in magnitude but are opposite

in sign. This is because of the direction of flowing electrons through the electric current.

Therefore, rearranging Eq.3.16, the cathodic and anodic activation polarization can be

determined as follows,

⎟⎟⎠

⎞⎜⎜⎝

⎛= −

coecAct i

iFn

RT

,

1, 2

sinh2η (3.17a)

⎟⎟⎠

⎞⎜⎜⎝

⎛= −

aoeaAct i

iFn

RT

,

1, 2

sinh2η (3.17b)

where ηAct,c and ηAct,a are the cathodic activation polarization and the anodic activation

polarization respectively, and io,c and io,a are the cathodic exchange current density and

anodic exchange current density respectively.

41

3.2.2 Ohmic Polarization

Resistance to the flow of ions through the electrolyte and resistance to the flow of

electrons between electrodes via electric circuit cause ohmic losses in the fuel cell.

These resistances obey Ohm’s law. Therefore, ohmic polarization can be expressed by

the equation,

eOhm Ri ⋅=η (3.18)

Re is the resistance of each material used in the fuel cell components and can be

calculated by the equation,

ρ⋅= wRe (3.19)

where w is the thickness and ρ is the electrical resistivity of each component.

3.2.3 Concentration Polarization

Certain changes in the concentration of the potential determining species ( i.e., ions )

will occur after current begins to flow in an electrochemical cell. This concentration

produces an electromotive force, which reduces the reversible electrical voltage of the

cell.

The concentration polarization is the reduction in potential due to a concentration

change of the electrolyte during a reaction in the vicinity of an electrode. Assuming that

the supply of the potential determining species to the electrode is by diffusion only,

Fick’s law of diffusion can be used to give the rate of diffusion. If the concentration of

species at the electrode is Ce and the bulk concentration is Co, then the rate of diffusion

can be written as [13],

( )nFICCDJ eo =−=

δ (3.20)

J is the mass flux ( i.e. mass transfer rate ), D is the diffusion coefficient, and δ is the

thickness of the diffusion layer. The diffusion layer is represented in Figure 3.3.

42

From Eq.3.20, it can be examined that when the electrolyte near the electrode is depleted

( i.e. Ce = 0 ), the maximum current occurs,

oL CnFDIδ

= (3.21)

IL is called the limiting current density. Limiting current is proportional to the bulk

concentration of the reactant. Assuming that the diffusion layer thickness is independent

of the rate of diffusion, combining Eqs.3.20 and 3.21 gives,

Lo

e

II

CC

−=1 (3.22)

Distance from electrode

Con

cent

ratio

n of

tran

sfer

red

spec

ies

True concentration profile

Equivalent concentration profile

Diffusion

δ

Ce

Co

Figure 3.3: The film thickness theory

43

Assuming that migration of the potential determining ion is negligible and that the ions

in the bulk fluid and near the electrode have the same activity coefficients, the

concentration polarization can be written as, [14]

⎟⎟⎠

⎞⎜⎜⎝

⎛=

o

eConc C

CnFRT lnη (3.23)

Concentration polarization can be written in terms of the limiting current density as

follows,

⎟⎟⎠

⎞⎜⎜⎝

⎛ −=

L

LConc I

IInFRT lnη (3.24)

Eq.3.24 refers to the electrode process where species is being removed from the

electrolyte. Concentration polarization at the opposite electrode where species is

generated and concentration builds up can be written as,

⎟⎟⎠

⎞⎜⎜⎝

⎛ +=

L

LConc I

IInFRT lnη (3.25)

From Eqs.3.24 and 3.25, it can be noticed that to calculate the concentration

polarization, the limiting current density must be calculated. In order to eliminate this

difficulty, another calculation method for the concentration polarization analysis may be

developed from the same point of view. Since the reactants and products are in gaseous

states, and a change in the partial pressure of the potential determining gaseous species

at the reaction zone occurs in respect to its partial pressure in the bulk of the gaseous

phase, gas side concentration polarization arises.

The relation of the limiting current density to the concentration is given in Eq.3.22 as,

Lo

e

II

CC

−=1 (3.22)

A similar relation may be estimated between the limiting current and the change in the

pressure of the fuel gas as follows.

44

We postulate a limiting current density, IL, at which the fuel is used up at a rate equal to

its maximum supply speed. The current density cannot rise above this value, because the

fuel gas cannot be supplied at a greater rate. At this current density the pressure will

have just reached zero [4].

If P1 is the pressure when the current density is zero, and assuming a linear drop of

pressure to zero, when the limiting current density is reached, then the pressure ratio of

P2, at any current density to P1 can be expressed as,

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

LII

PP 1

1

2 (3.26)

Substituting Eq.3.26 into Eq.3.24, the following relation is obtained.

⎟⎟⎠

⎞⎜⎜⎝

⎛=

1

2lnPP

nFRT

Concη (3.27)

Eq.3.26 is named as the gas side concentration polarization.

The gases have to diffuse through the gas filled pores of the electrode in order to reach

the reaction sites. Whenever this happens, it is possible for the concentration

polarization to be significant, since when the current is being drawn, the gas partial

pressure at the reaction sites will be less than that in the bulk of the gas stream. Hence,

decrease of gas concentration in the gas filled pores of the electrode may give rise to

severe polarization and lead to limiting current. [15]

To determine the concentration polarization, the mass transport effects in the fuel cell is

to be investigated. The concentration polarizations at anode and cathode are obtained in

the next section.

45

3.3 Mass Transport Effects

Diffusion through the porous material is described by molecular diffusion, Knudsen

diffusion, or transition region diffusion ( the case both molecular and Knudsen diffusion

occurs ). Since the pores of porous solids are usually very small, the diffusion of gases

depends on the diameter of the pores. Because of this reason, different diffusion types

may occur. To determine which mechanisms of diffusion occur, a mean free path is

defined firstly. The mean free path is the average distance a molecule passes through

until it collides with another molecule. Mean free path is given as follows [16].

MgRT

P cπµλ

22.3

= (3.28)

In the equation, λ is the mean free path, µ is the viscosity, gc is the gravitational

acceleration, and M is the molecular weight.

After the calculation of the mean free path, from the comparison between the mean free

path and the average pore diameter, the diffusion type can be determined. Each diffusion

type is shown in Figure 3.4a – c [16].

In the Figures 3.4a - c, NA indicates the flux of the molecules A, pA,I and pA,o are the

total pressure at the inlet and exit of the pore. The total pressure on either side in these

cases is the same, but the partial pressures of the molecules may be different.

3.3.1 Knudsen Diffusion

As shown in Figure 3.4a, if the mean free path is very large compared to the average

pore diameter, the molecule collides with the pore walls frequently. This type of

diffusion is called Knudsen diffusion. The Knudsen diffusion coefficient is given as

follows, [16]

46

λ

NA

d

PA,i PA,o( a )

λ

47

Figure 3.4: Types of diffusion through the pores: ( a ) Knudsen diffusion, ( b ) molecular diffusion, ( c ) transition diffusion. [16]

d

( b )

λ

NA

PA,i PA,o

NA

d

PA,i PA,o( c )

AKA M

TrD ⋅⋅⋅= −3107.9 (3.29)

DKA is the Knudsen diffusion coefficient of A, r is the average pore radius.

3.3.2 Molecular Diffusion

Figure 3.4b shows the mechanism of molecular diffusion. If the average pore diameter is

very large compared to the mean free path, then molecular diffusion, where molecules

collide with each other frequently, occurs. Due to this collision of molecules, the

molecular diffusion coefficient of a binary gas pair A-B, using the Chapman – Enskog

theory of prediction, can be given as, [16]

ABDAB

D

BAAB P

fTMM

D,

2

2321110018583.0

Ω⎟⎟⎠

⎞⎜⎜⎝

⎛+=

σ (3.30)

DAB is the diffusion coefficient of gas A diffusing in gas B, σAB is the collision diameter,

ΩD,AB is the collision integral based on the Lennard-Jones potential, which is obtained

from the energy of molecular interaction, eAB, and fD is a small second order correction

factor, which is close to 1 and may be dropped.†

3.3.3 Transition Region Diffusion

If neither the mean free path, nor the average pore diameter is very large compared to

the other, then transition or mixed type diffusion occurs, as shown in Figure 3.4c. In this

diffusion regime, both molecule to molecule collisions and molecule to wall collisions

become significant, and they both must be taken into account. Since both molecular

diffusion and Knudsen diffusion may occur simultaneously, diffusion coefficient for the

transition region may be written as,

48

† 2

BAAB

σσσ += and

ke

ke

ke BAAB ⋅= are calculated from the relational tables.

KAABtA DDD111

+= (3.31)

DtA is the diffusion coefficient of gas A in transition region.

Since the diffusion of molecules will be through tortuous path in the pores of the

electrodes, the two factors “porosity” and “tortuosity” are taken into account and

therefore the effective diffusion coefficients for each diffusion type is given as follows,

( ) KAeffKA DDξβ

= (3.32a)

( ) ABeffAB DDξβ

= (3.32b)

( ) ( ) ( )effKAeffABKAABefftA DDDDD11111

+=⎟⎟⎠

⎞⎜⎜⎝

⎛+=

βξ (3.32c)

In the Eqs.3.32a – c, β is the porosity, and ξ is the tortuosity.

Now the anode and cathode electrode reactions can be examined. By using the diffusion

coefficients, the concentration polarization can be calculated for both parts.

On the anode side, counter-current equimolar diffusion of the reactant ( H2 for this case )

and the product ( H2O for this case ) occur. Therefore, the molar flux for the anode side

can be written as [17],

( ) dxdCDJ A

effAA −= (3.33)

The subscript A is H2 or H2O ( i.e., reactant or product ), and D represents the suitable

diffusion type occurring in the anode.

For the reactant, H2, using Eq.3.20, with n = 2, since 1 mole of H2 reacts with 2

electrons, and from the knowledge of the ideal gas equation of state, the following

equations can be obtained.

⎟⎠⎞

⎜⎝⎛−=

FiJ H 22

(3.34)

49

⎟⎟⎠

⎞⎜⎜⎝

⎛=

RTdp

dC HH

2

2 (3.35)

Substituting Eqs.3.34 and 3.35 into Eq.3.33 and solving for the pressure, for the

hydrogen the pressure change can be written as,

( )dx

FDRTidp

effHH

2

2 2= (3.36)

Integrating this equation gives,

( )a

effHH

IH l

FDRTipp

2

22 2=− (3.37)

The superscript “I” is used to denote initial values of the pressure of the hydrogen, la is

the thickness of the anode.

For the product, H2O, using the same equations as for the hydrogen, the following

reaction is obtained.

( )a

effOH

IOHOH l

FDRTipp

2

22 2=− (3.38)

Rearranging the Eqs.3.37 and 3.38, partial pressures of hydrogen and water vapor in

terms of current density are obtained as follows,

( )i

FDRTlpp

effH

aIHH

2

22 2−= (3.39)

( )i

FDRTlpp

effOH

aIOHOH

2

22 2+= (3.40)

At the cathode side, on the other hand, there are two gases, and diffusion is

nonequimolar. The molar flux for this case can be written as,

( ) JXdx

dCDJ AA

effAA +⎟⎠⎞

⎜⎝⎛−= (3.41)

50

J = JA + JB is the total flux, where subscripts A and B are the two diffusing components,

XA is the molar ratio of component A. In terms of the molecular diffusion coefficient,

Eq.3.41 becomes,

( ) JXdx

dCDJ AA

effMAA +⎟⎠⎞

⎜⎝⎛−= (3.42)

Subscript “MA” is used to indicate molecular diffusion coefficient of gas A, in order not

to confuse with DA. The flux at the cathode in terms of Knudsen diffusion coefficient is

given as,

( ) ⎟⎠⎞

⎜⎝⎛−=

dxdCDJ A

effKAA (3.43)

Solving Eqs.3.42 and 3.43 for the concentration drop terms gives,

( ) ( ) ( )J

DX

JDDdx

dC

effMA

AA

effKAeffAM

A −⎟⎟⎠

⎞⎜⎜⎝

⎛+=⎟

⎠⎞

⎜⎝⎛ 11 (3.44)

Thus, flux becomes,

( ) ( ) ⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛+

+

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛+

−=

)()()(

11

1

11

1

effKAeffMAeffMA

AA

effKAeffMA

A

DDD

JXdx

dC

DD

J (3.45)

The flux can be written in a simpler equation as follows,

( ) AAA

effAA JXdx

dCDJ δ+−= (3.46)

δ is defined as,

( )

( ) )(effMAeffKA

effKAA DD

D+

=δ (3.47)

If the molecular diffusion coefficient is very large compared to the Knudsen diffusion

coefficient, then DA (eff) = DKA (eff) , and δ approaches zero. If the Knudsen diffusion

51

coefficient is very large compared to the molecular diffusion coefficient, then

DA (eff) = DKA (eff) , and δ = 1.

At the cathode, since the flux of nitrogen is zero, using the Eq.3.46, the flux of oxygen is

given as,

222

2

22 )( OOOO

effOO JXdx

dCDJ δ+−= (3.48)

( )

( ) )(22

2

2effMOeffKO

effKOO DD

D+

=δ (3.49)

For oxygen, using Eq.3.20, with n = 4, since one mole of oxygen produces 4 electrons,

and from the knowledge of the ideal gas equation of state, the following equations can

be obtained.

FiJO 42

−= (3.50)

RTdp

dC OO

2

2= (3.51)

Substituting Eqs.3.50 and 3.51 into Eq.3.48 gives,

( ) dxFD

RTiXdp

effOOO

O

)(222

2

41=

− δ (3.52)

Since the partial pressure of oxygen is equal to its molar ratio at the cathode part, each

terms in the Eq.3.52 can be divided by the cathode pressure, pc. Hence,

( ) dxpFD

RTipp

dp

ceffOOOc

O

)(222

2

4=

− δ (3.53)

Integrating Eq.3.53 gives,

(3.54) cceffO

O

IO

O

c

OO

c

lpFD

RTi

pp

pp

)(2

2

2

2

2

2

4ln

δ

δ

δ=

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

−⎟⎟⎠

⎞⎜⎜⎝

⎛

−⎟⎟⎠

⎞⎜⎜⎝

⎛

52

The partial pressure of the oxygen at the cathode, therefore, will be,

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−−=

ceffO

cOIO

O

c

O

cO pFD

RTilp

ppp

)(2

2

2

22

2 4exp

δδδ

(3.55)

The partial pressures of the gases at both the anode and cathode sides are obtained.

Substituting the obtained partial pressures ( i.e. Eqs.3.39 , 3.40, and 3.55 ) into Eq.3.27,

the concentration polarization at the anode is given by,

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛

−=

IOH

OH

IH

H

aConc

pp

pp

FRT

2

2

2

2

ln2,η (3.56a)

Hence, the concentration polarization at the anode side is obtained as follows.

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛+

⎟⎟⎠

⎞⎜⎜⎝

⎛−

−=

ipFD

RTl

ipFD

RTl

FRT

IOHeffOH

a

IHeffH

a

aConc

22

22

)(

)(,

21

21

ln2

η (3.56b)

Concentration polarization at the cathode, using the same way, is given,

⎟⎟⎠

⎞⎜⎜⎝

⎛−= I

O

OcConc p

pF

RT

2

2ln4,η (3.57a)

Therefore, the concentration polarization at the cathode is,

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−−

−= IO

ceffO

cOIO

O

c

O

c

cConc p

ipFD

RTlppp

FRT

2

2

2

2

22 )(,

4exp

ln4

δδδ

η (3.57b)

53

54

CHAPTER 4

MODELING AND CALCULATION

4.1 Fuel Cell Type Selection

As a first step, the type of fuel cell that will be studied on throughout the study must be

selected. Solid oxide fuel cell with external reformer is selected for the calculations.

Solid oxide type fuel cell has the following advantages:

• SOFC unit sizes can be as small as 2.5 kW, and as large as 100 kW.

• Its ability to be used for power production applications.

• SOFC is simple in concept than the other types of fuel cell systems, since it is a

completely solid state device that uses an oxide ion conducting ceramic

electrolyte, whereas the other types require solid and gas phases.

• The solid state character of the SOFC components means that there is no

restiriction on the cell configuration, in principle. SOFC can be designed as

tubular or as flat plate.

The material data of the SOFC type selected referencing from [18], and is given in Table

4.1.

Table 4.1: Properties of SOFC materials Component

Anode Cathode Electrolyte InterconnectorMaterial LSC YSZ

Thickness ( µm ) 150 1000 100 100 Resistivity ( Ω cm )

at 1073 K 0.0003 0.001 17.24 0.033 at 1273 K 0.0003 0.0011 10 0.033

3LaCrO2/ ZrONi

4.2 Environment and Air Composition

As mentioned in chapter 2, the second law analysis requires that the environment is

defined. For the study, we assume the environment to be in standard state

( i.e. To = 298.25 K, Po = 1 atm ).

The air is assumed to be consist of oxygen and nitrogen only. The molar ratios of

oxygen and nitrogen are 20% and 80% respectively.

4.3 Chemical Reactions and Components of SOFC System

Methane is used as the fuel for the SOFC system. Direct oxidation of methane has not

been successful in calculations so far, because carbon that is a product of oxidized

methane is an unwanted element at the anode part since it causes carbonization. Current

technology of electrodes are not sufficient to produce such an anode to avoid this

problem.

For the simulation, SOFC system with external reformer will be used. Hydrogen will be

produced by reforming methane with water vapor. Therefore, the stoichiometric reaction

of methane with water vapor will be as follows,

55

2224 42 COHOHCH +→+ ( 4.1 )

As mentioned earlier, carbon is an unwanted element, since it causes carbonization at the

anode. To avoid this problem, additional moles of water vapor are used to prevent

possible thermal decomposition of the methane that would otherwise form solid carbon

which would deactivate the catalytic reaction of the reformer [12]. From literature

survey, water – methane mole ratios of 3.0 and 2.2 seem to be conservative values [11].

Selecting water – methane mole ratio to be 2.2, the stoichiometric reaction in Eq.4.1 can

be given as,

OHCOHOHCH 22224 2.042.2 ++→+ ( 4.2 )

The water vapor that is required to reform methane to produce hydrogen is supplied to

the system after vaporized. This reaction in the vaporizer is simply given as

( ) ( )gOHlOH 22 → ( 4.3 )

Temperature is kept constant. This reaction, since it is endothermic ( i.e. requires heat )

is heated by the afterburner.

After vaporizer, water vapor is mixed with methane in the mixer and then before

reformer reaction, they are heated by the pre-heater. This is because SOFC unit requires

high operating temperature.

In the SOFC, the reactions at the anode and cathode will be as follows;

(cathodeOeO −− →+ 22 2½ )

)

(4.4a)

( ) (anodeOcathodeO −− → 22 (4.4b)

( )anodeeOHOH −− +→+ 222

2 (4.4c)

Hence, the overall reaction in the cell is,

OHOH 222 ½ →+ (4.4d)

56

The degree of fuel utilization in the cell unit has a very strong effect on the efficiency of

the system. From the literature survey, it is found that if the supplied fuel is hydrogen

then it may have a high fuel utilization rate almost very close to the ideal case

( i.e. 100% ). If the supplied fuel is a gas mixture, fuel utilization rate may change

between 0.7 – 0.85.

Conversion rates of hydrogen and oxygen are interrelated. Oxygen ions ( O2- ) are

assumed to be always available at the anode for hydrogen to react to produce water

vapor.

The reformer reaction is endothermic. The required heat is supplied by the afterburner,

where the unreformed methane and surplus hydrogen are burnt. The reactions in the

afterburner are assumed to be complete. Afterburner is also assumed to be isothermal

( i.e. temperature is the same at the inlet and exit of the afterburner ). The reactions in

the afterburner are as follows;

2224 22 aCOOaHaOaCH +→+ (4.5a)

ObHbObH 222 ½ →+ (4.5b)

The variables “a” and “b” are used to indicate the unreformed methane and unreacted

hydrogen mole numbers.

In the foregoing discussion, the composition of air is defined. Regarding to the fuel cell

unit and the afterburner reactions, the amount of oxygen, therefore the amount of air,

required can be calculated. This amount of air, before entering the SOFC unit, is heated

by the pre-heater, since SOFC requires high operating temperature.

The pre-heaters that have been discussed use the waste heat of the products to heat the

fuel and air inlet parts. For the design of the pre-heaters, heat exchanger method is

explained using the ε-NTU method. The counter flow type heat exchangers are selected,

and the total heat transfer coefficients are set equal to 0.05 kW/m².

57

4.4 Simulation Models

4.4.1 Simulation Model 1: SOFC Unit Analysis

The first model is used to investigate the fuel cell and reformer reactions, and their

effects on the SOFC second law efficiency. Besides, the effects of polarizations on fuel

cell voltage and power density are also examined. Each polarization type effects and

their change with current density are examined and are presented in graphs. This model

is limited to the calculation of a unit cell, operating at constant temperature and pressure

only. Operating the fuel cell unit with different temperature and pressure values, its

second law efficiency change and polarization changes are examined. The schematic of

this model is given in Figure 4.1.

58

Figure 4.1: Schematic of simulation model 1.

SOFC Stack Reformer

3

4 1

2

4.4.2 Simulation Model 2: SOFC System Analysis

The second model is more complex than the first one. The components: afterburner, two

preheaters, mixer and a vaporizer, are added in addition to the basic system requirements

described for the first model. The model of the system is given in Figure 4.2.

59

Figure 4.2: Schematic of simulation model 2.

SOFC

Reactor

Vaporizer

Preheater 2

Mixer

Preheater 1

Reformer

Afterburner

11

2 7

Wact Air Inlet

Qlost

10

1 4 5

6

3 9

8Q1

Q2

Qo

Environment, To

Water

Fuel, water vapor and air inlet conditions are taken to be at standard temperature and

pressure conditions. Water vapor, being vaporized in the vaporizer and methane enter

the mixer and after mixing, they are heated in preheater, before entering the reformer

( the mixer component is not shown in Figure 4.1. The fuel and water vapor entering the

reformer are mixed ). The heat required for the vaporizer and the reformer is supplied by

the afterburner. Some heat is also lost to environment from the afterburner.

The fuel cell fuel inlet temperature, and hence the reformer exit temperature will be

assumed to be known initially, and so will the fuel cell air inlet temperature be. The

SOFC requires high operating temperature. Because of this reason, the assumed

temperatures will be high ( around 800 – 1000°C.). Theoretical percentage of the air,

fuel utilization rate and the reformer efficiency are also assumed to be known. The

variables to be assumed will be discussed later.

4.5 Electrochemical Model

As mentioned earlier, three types of polarization occur in the cell. Activation and

concentration polarizations take place in the anode and cathode sides, while the ohmic

polarization takes place in all of the components of the fuel cell ( i.e. electrolyte, anode,

cathode, and interconnector ). Using the relative equations derived in sections 3.2 and

3.3, the polarization values can be calculated. As a summary, the polarization equations

needed to calculate the voltage drop in the cell are given below.

⎟⎟⎠

⎞⎜⎜⎝

⎛= −

aoeaAct i

iFn

RT

,

1, 2

sinh2η (3.16b)

⎟⎟⎠

⎞⎜⎜⎝

⎛= −

coecAct i

iFn

RT

,

1, 2

sinh2η (3.16a)

RiOhm ⋅=η (3.17)

60

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛+

⎟⎟⎠

⎞⎜⎜⎝

⎛−

−=

ipFD

RTl

ipFD

RTl

FRT

IOHeffOH

a

IHeffH

a

aConc

22

22

)(

)(,

21

21

ln2

η (3.55b)

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−−

−= IO

ceffO

cOIO

O

c

O

c

cConc p

ipFD

RTlppp

FRT

2

2

2

2

22 )(,

4exp

ln4

δδδ

η (3.56b)

After each of the polarizations is calculated, using Eq.3.1, the actual electric voltage can

be determined.

( )cConcaConcOhmcActaActoEE ,,,, ηηηηη ++++−= ( 4.6 )

Eo is the Nernst potential and is given by the Eq.2.68a and 2.68b.

Let us consider the following reaction.

OHOH 222 ½ →+ ( 4.7 )

The Nernst potential for the Eq.4.7 is given as follows.

( )( )( ) ⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛+=

OH

IO

IH

o ppp

FRTK

FRTE

2

22

½

ln2

ln2

( 4.8 )

Once the polarizations and the Nernst potential are determined, the actual cell voltage

can be calculated. The electrical power density produced by the fuel cell is calculated by

the following equation.

IEP ⋅= ( 4.9 )

For the fuel cell reaction of the control volume in Figure 2.1, assuming steady state

steady flow and negligible changes in kinetic and potential energies, the energy balance

in terms of molar properties and molar flow rates can be written as,

61

( ) ( )∑ ∑−=−i i

RiiPii hnhnWQ &&&& (4.10)

For the same reaction, the entropy balance can be written as,

( ) ( )∑ ∑ −−=i i

genRiiPii SsnsnTQ &&&&

(4.11)

Combining Eqs.4.10 and 4.11, the exergy balance can be obtained.

( ) ( ) geni

Riiiii

Piiii STsnThnsnThnW &&&&&& +−−−=− ∑∑

(4.12)

The work done by the cell is the electrical work. Hence, from Eq.2.61, Eq.4.12 can be

rearranged as follows.

( ) ( ) geni i

RiiPii STgngnnFE &&& −+−= ∑ ∑ (4.13)

Substituting Eqs.2.64 and 2.65 into Eq.4.13 gives,

geni Ro

Ii

iPi o

ii

o STpp

RTnpp

RTnGnFE &&& −⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛−∆−= ∑∑ lnln (4.14)

Substituting Eq.2.62 into Eq.4.14, and solving for the cell voltage yields,

( )( )( )( ) nF

ST

pp

ppnFRTK

nFRTE gen

IO

IH

oOH&

−⎟⎟

⎠

⎞

⎜⎜

⎝

⎛−= ½

½

22

2lnln (4.15)

The first term on the right hand side of the Eq.4.15 shows the effect of temperature on

the cell voltage, the second term shows the effect of the pressures on the cell voltage.

The third term shows the irreversibility drop ( i.e. polarization ) in the cell voltage.

The heat lost to the environment from the fuel cell can be calculated from,

( )genSSTQ && −∆= (4.16)

62

The terms ∆S and are defined as follows. genS&

( ) ( ) ⎟⎟

⎠

⎞

⎜⎜

⎝

⎛−−−=∆ ½

22

2

222ln½

OH

OHoO

oH

oOH pp

pRSSSS (4.17)

( cConcaConcOhmcActaActgen TnFS ,,,, ηηηηη ++++=& ) (4.18)

4.6 Heat Exchanger Model

To model the preheaters of the SOFC system ( i.e. simulation model 1 ), effectiveness-

NTU method ( ε-NTU ) is used. The counter flow type heat exchangers are selected as

the preheaters. The heat transfer coefficients are set equal to 0.05 kW/m²K. The areas of

the heat exchangers must be determined before the calculation steps, in order to use the

effectiveness-NTU method. Because of this reason, the preheaters are first sized

according to the assumed operating conditions. After sizing, the performance analysis

can be carried out.

The outlet temperatures of the preheaters depend on the inlet conditions. To define the

effectiveness of the heat exchanger, the maximum possible heat transfer rate, qmax must

be determined. This maximum possible heat transfer rate depends on the minimum heat

capacity rate. [19] To determine the minimum heat capacity rate, the heat capacity rates

of the cold and hot gas streams are calculated from the following equations.

∑= cpcc cmC ,& (4.19a)

∑= hphh cmC ,& (4.19b)

The subscripts “c” and “h” indicate cold and hot gas streams respectively. Comparing

the heat capacity rates of the cold and hot gas streams, one with the lower value is

determined as the minimum heat capacity rate, while the other is determined as the

maximum heat capacity rate. Following this, the heat capacity rates ratio is given as,

63

max

min

CCCr = (4.20)

Hence, the maximum possible heat transfer rate can be calculated by,

( )icih TTCq ,,minmax −= (4.21)

Th,i is the inlet temperature of the hot gas stream, and Tc,i is the inlet temperature of the

cold gas stream.

The number of transfer units ( NTU ) is a dimensionless parameter that is used for heat

exchanger analysis and is defined as,

minCUANTU = (4.22)

U is the heat transfer coefficient; A is the area of the heat exchanger.

The effectiveness of the counter flow type heat exchangers can be calculated from the

following equation.

( )[ ]([ )]rr

r

CNTUCCNTU−−−

−−−=

1exp11exp1

ε (4.23)

The effectiveness, ε, is defined as the ratio of the actual heat transfer rate to the

maximum possible heat transfer rate. Therefore, with known effectiveness and

maximum possible heat transfer rate, the actual heat transfer rate can be calculated.

maxqq ⋅= ε (4.24)

The energy balance for the cold and hot gas streams can be written as follows.

( )icocc TTCq ,, −= (4.25a)

( )ohihh TTCq ,, −= (4.25b)

Tc,o and Th,o represents the outlet temperature of the cold gas stream and the outlet

temperature of the hot gas stream respectively. As a result, the cold and hot gas streams’

outlet temperatures can be found by rearranging Eqs.4.25a – b, as follows.

64

cicoc C

qTT += ,, (4.26a)

hihoh C

qTT −= ,, (4.26b)

The hot gas stream inlet temperature is also the SOFC outlet temperature. Because of

this, the hot gas inlet temperatures of both of the preheaters are affected by the SOFC

outlet temperature. For this reason, the calculation steps for the heat exchanger model

are repeated until the outlet temperatures of the gas streams from the preheaters

converge.

4.7 Calculation Procedure

In the calculations, two different models are simulated as mentioned. The reason for this

is first of all to investigate the effects on the fuel cell unit and the changes of effects with

different conditions. After this study, using a simple model, a general thermodynamic

analysis for every component of the fuel cell system is made.

The model 1 is simulated in order to examine

• The effect of reformer efficiency and fuel utilization rate on the fuel cell second

law efficiency,

• The effect of pressure on the fuel cell second law efficiency,

• The comparison of the results obtained with different temperatures,

• The polarization effects on fuel cell voltage,

• The effect of temperature on fuel cell voltage and power density.

On the other hand, the more complex system, model 2, is simulated in order to make a

full thermodynamic analysis on each component of a simple SOFC system. The first

model deals with the SOFC unit with external reformer only. But the second model

examines the use of the waste heat at the exit of the SOFC unit to heat the reformer and

65

66

vaporizer for the reactions to take place, and to heat the fuel and air inlet nodes to the

required temperatures. For these purposes afterburner and two preheaters are added to

the system. The preheaters are designed using the heat exchanger model.

4.7.1 General Assumptions

Before the calculation steps, the general assumptions and conditions for the simulation

models are listed as follows;

• Steady flow throughout the nodes,

• Kinetic and potential energy changes are neglected,

• Frictional effects are assumed to be negligible,

• Fuel supplying rate is 1 kg/s,

• Operating temperature is constant throughout the fuel cell unit ( i.e. fuel cell unit

inlet, exit and reformer inlet temperatures are all the same ) ( for model 1 only ),

• Afterburner is assumed to be isothermal, i.e. the inlet and outlet temperatures are

the same ( for model 2 only ),

• The reactions in the afterburner are assumed to be complete ( for model 2 only),

• The environment is at standard temperature and pressure conditions, i.e. 298 K,

and 1 atm.

The calculations are carried out under the given assumptions. Since simulation model 1

will investigate the effects of the reformer efficiency, fuel utilization rate, the operating

pressure on the fuel cell efficiency and the effects of polarizations on fuel cell voltage

and power density with changing current density, there is no need to make additional

assumptions for these variables for model 1. Hence, the general assumptions are applied

to the simulation model 1 directly. For simulation model 2, the following variables will

be assumed to be known initially, and the values will be given for these variables.

• The operating pressure of the fuel cell system,

• Percentage of theoretical air,

• Reformer efficiency,

• Fuel utilization rate,

• Fuel cell fuel inlet and air inlet temperatures.

4.7.2 Calculation Steps for Simulation Model 1

Before thermodynamic analysis of the fuel cell unit, using the electrochemical model,

the voltage drops in fuel cell unit due to polarizations, their effect on fuel cell voltage

and power density are calculated.

The model is first assumed to be at T = 1273 K , and then at T = 1073 K. The

electrochemical model to calculate the polarizations is used for both temperature values.

By this way, the effect of temperature on fuel cell power density, voltage and

polarizations can be considered. The anode and cathode exchange current densities are

taken as Io,a = 70000 A / m² , Io,c = 24000 A / m² for T = 1273 K, and Io,a = 5000 A / m²,

Io,c = 2000 A / m² for T = 1073 K [12].

After the calculation of the polarization effects using the electrochemical model,

thermodynamic analysis of the fuel cell unit is made.

Exergy efficiency is defined as the ratio of the work output to the maximum work output

of the system. Hence, it can be written as follows,

max

max

WWW lost

II−

=η (4.27)

The maximum work available for the system is given by the Eq.2.45. Writing Eq.2.45 in

terms of molar properties,

STHGW o∆−∆=∆=max (4.28)

The work losses are due to incomplete reactions in the reformer and fuel cell unit, and

polarizations occurring in the fuel cell unit. Since hydrogen is reformed from methane

67

gas, the moles of hydrogen depend on reformed methane, and so do the moles of

oxygen. The reactions occurring in reformer and at the anode and cathode of the fuel cell

are written assuming methane is completely reformed and the fuel cell reactions are

ideal.

( )reformerOHCOHOHCH 22224 2.042.2 ++→+ (4.29a)

( )anodeeOHOH −− +→+ 8444 22

2 (4.29b)

( )cathodeOeO −− →+ 22 482 (4.29c)

The overall reaction of the system is the sum of reactions 4.17a – c.

2224 22 COOHOCH +→+ (4.30)

The maximum work of the system is calculated using complete conversion ratios

( i.e. 14=CHη , 1

2=Hη , and 1

2=Oη ).§ In reality, maximum values of the conversion

ratios will not be unity, but smaller. These differences cause incomplete reactions and

hence work loss.

The reformer reaction, since it is an irreversible combustion process, is a complete loss

of work. Hence work loss can be calculated using the Gibbs function.

STHGW refloss ∆−∆=∆=, (2.45)

The work loss for the incomplete reactions can be calculated the same way, since the

conversion ratio decreases the available work output of the reaction.

STHGW incloss ∆−∆=∆=., (2.45)

In the electrolyte cracks may occur. This means that the cell is short circuited, some fuel

reacts irreversible with oxygen and this produces hot spots. [20] An assumption of 1 %

of the hydrogen react in this way, the work loss due to cracks is given by the following

equation.

68

§ 224

,, OHCH ηηη are the conversion ratios of CH4, H2, and O2, respectively. This value indicates how much of CH4 is reformed and how much of H2, and O2 is reacted in the reaction. Since maximum work is the reversible work, the ideal conversion ratios are assumed, which is not the case realistically.

reactionHreactionHcrackloss STHGW −− ∆−∆=∆⋅=22

)(01.001.0, (4.31)

The polarizations occurring in the fuel cell causes work loss. From the electrochemical

model, the polarizations are calculated with current density. Hence, operating fuel cell

with an assumed current density, the work loss due to polarizations can be calculated.

The work loss caused by polarizations is,

69

)( OhmConcActirrloss FnW ηηη ++⋅⋅=, (4.32)

The number of electrons, n, is 8 in ideal case. But, that depends on the conversion ratios

of methane, hydrogen, and oxygen, and is calculated according to the given conversion

ratios.

As seen from Figure 4.1, there are four nodes in model 1. At each node of the system,

there is a mixture of gases, and since partial pressures apply, the composition of the

mixtures is to be figured out. For this reason, before analyzing the system

thermodynamically, the mixture compositions at each node of the system regarding to

the conversion ratios of each reaction are determined. With the mixture components

known at each node, the related properties of the gases can be calculated.

After thermodynamic analysis is accomplished, the heat release in the fuel cell will also

be examined.

4.7.3 Calculation Steps for Simulation Model 2

The complete reforming of methane in the reformer and complete reaction of hydrogen

in the fuel cell stack are not realistic.

Reforming of methane is an endothermic reaction and requires heat input to the system

for the reaction to take place. The vaporizing of liquid water to water vapor is also an

endothermic reaction and requires heat input. Because of these heat requirements of the

reformer and vaporizer reactions, there must be some fuel left at the exit of the fuel cell.

The afterburner is added to the exit of the fuel cell stack in order to burn the remaining

fuel and release heat required for the reformer and vaporizer reactions.

70

Before calculations, the heat requirements of the components are investigated.

Regarding to these results, the conversion ratios for the reformer and the fuel cell unit

reactions are determined.

Before stepping into calculations, using the heat exchanger model, the heat areas of the

two preheaters are calculated. The system is modeled stoichiometrically, that is the air

supplied to the system is 100% theoretical ( i.e. there is no extent of air ).

With the calculations explained above, the reformer efficiency due to the conversion

ratio of methane, the fuel utilization rate due to the conversion ratio of hydrogen are

determined, and the heat transfer area needed in the two preheaters are calculated.

Modeling the inlet conditions at standard temperature and pressure ( i.e. T = 298 K,

P = 1 atm ), under the general assumptions and the assumed and calculated values for the

reformer efficiency, fuel utilization rate and heat transfer area, the simulation model 2

can be analyzed. The calculation steps for this model are numbered and listed in order as

follows:

1. Since the methane and hydrogen conversion ratios are determined, and there is a

complete combustion in the afterburner, the required air amount by the SOFC

stack and the afterburner to sustain the system operation is calculated.

2. The molar chemical compositions of the flow streams at each node of the system

are determined.

3. Using the known molar chemical compositions of the flow streams at each node

and the fuel supplying rate, the mass flow rates at each node are determined.

4. The temperature values of the fuel and air inlet must be assumed. With known

fuel and air inlet temperatures, the electrochemical model can be used in order to

determine the electrical work output, the exit temperature, and the heat rejection

of the SOFC stack. The first law and second law efficiencies of the SOFC stack

are also calculated.

5. Heat exchanger model is used to determine the exit temperatures of the heat

exchangers. The reformer inlet temperature is calculated at this step. The cold

gas stream outlet temperature of the preheater 2, which is also the assumed air

71

inlet temperature of the SOFC, is compared with the calculated result. Since the

exit temperatures of the preheaters depend on the exit temperature of the SOFC

stack, the iteration is continued until the outlet temperature of the preheater 2

converges.

6. Since temperatures and molar chemical compositions at each node are known,

applying the energy and exergy balance at each component, the enthalpy and

exergy at each node are calculated.

The exergy balance for a component is derived in Eq.2.74 and is used to calculate the

exergy destruction in the component. The exergy of a mixture at a node, where the

temperature and molar chemical compositions are known, is calculated by using

Eqs.2.76 and 2.77.

72

CHAPTER 5

RESULTS AND DISCUSSION

5.1 Results of Simulation Model 1

5.1.1 Electrochemical Model Analysis

The first electrochemical results for the simulation model 1 are calculated at T = 1273 K.

The changes in activation, ohmic, and concentration polarizations with current density

are given in Figures 5.1, 5.2, and 5.3, respectively. The change in cell voltage and power

density due to the polarizations is given in Figure 5.4. The change in all kinds of

polarizations with current density is grouped and shown in a single graph in Figure 5.5.

The calculated power density, cell voltage, and polarizations versus current density is

given in Figure 5.6.

0

0,01

0,02

0,03

0,04

0,05

0 2000 4000 6000 8000 10000 12000

Current Dens ity ( A / m ² )

Act

ivat

ion

Pola

rizat

ion

( V )

Anode Cathode

T = 1273 K

Figure 5.1: Activation polarization change with current density ( T = 1273 K ).

0

0,01

0,02

0,03

0,04

0,05

0,06

0 2000 4000 6000 8000 10000 12000

Current Density ( A / m ² )

Ohm

ic P

olar

izat

ion

( V )

T = 1273 K

Figure 5.2: Ohmic polarization change with current density ( T = 1273 K ).

73

0

0,05

0,1

0,15

0,2

0,25

0,3

0 2000 4000 6000 8000 10000 12000Current Density ( A / m² )

Con

cent

ratio

n Po

lariz

atio

n ( V

)

Anode Cathode

T = 1273 K

Figure 5.3: Concentration polarization change with current density ( T = 1273 K ).

0

0,2

0,4

0,6

0,8

1

1,2

0 2000 4000 6000 8000 10000 12000


Cel

l Vol

tage

( V

)

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

Pow

er D

ensi

ty (

W /

m² )

Cell Voltage Pow er Density

T = 1273 K

Figure 5.4: Change in the cell voltage and the power density with current density ( T = 1273 K ).

74

0

0,01

0,02

0,03

0,04

0,05

0,06

0,07

0,08

0,090,

1

020

0040

0060

0080

0010

000

1200

0

Cur

rent

Den

sity

( A

/ m² )

Voltage ( V )

Anod

e Ac

tivat

ion

Pola

rizat

ion

Cat

hode

Act

ivatio

n Po

lariz

atio

nO

hmic

Pol

ariza

tion

Anod

e C

once

ntra

tion

Pola

rizat

ion

Cat

hode

Con

cent

ratio

n Po

lariz

atio

n

T =

1273

K

Figu

re 5

.5: C

alcu

late

d po

lariz

atio

n ef

fect

s with

cur

rent

den

sity

( T

= 12

73 K

).

75

0

0,2

0,4

0,6

0,81

1,2

020

0040

0060

0080

0010

000

1200

0Cu

rren

t Den

sity

( A

/ m² )

Cell Voltage ( V )

01000

2000

3000

4000

5000

6000

7000

8000

9000

1000

0

Power Density ( W / m² )

Ano

de A

ctiv

atio

n Po

lariz

atio

nCa

thod

e A

ctiv

atio

n Po

lariz

atio

nO

hmic

Pol

ariz

atio

n

Ano

de C

once

ntra

tion

Pola

rizat

ion

Cath

ode

Conc

entra

tion

Pola

rizat

ion

Cell V

olta

ge

Pow

er D

ensi

ty

T =

1273

K

Figu

re 5

.6: C

alcu

late

d po

wer

den

sity

, cel

l vol

tage

, and

pol

ariz

atio

ns w

ith c

urre

nt d

ensi

ty

( T =

127

3 K

).

76

For a second condition, temperature is decreased to T = 1073 K. The same results

graphed for T = 1073 this time.

The changes in activation, ohmic, and concentration polarizations with current density

are given in Figures 5.7, 5.8, and 5.9. The change in cell voltage and the power density

due to the polarizations is given in Figure 5.10. The change in all kinds of polarizations

with current density is grouped and shown in a single graph in Figure 5.11. The

calculated power density, cell voltage, and polarizations versus current density is shown

in Figure 5.12.

0

0,05

0,1

0,15

0,2

0,25

0,3

0,35

0 2000 4000 6000 8000 10000 12000


Act

ivat

ion

Pola

rizat

ion

( V )

Anode Cathode

T = 1073

Figure 5.7: Activation polarization change with current density ( T = 1073 K ).

77

Ohmic

0

0,04

0,08

0,12

0,16

0,2

0 2000 4000 6000 8000 10000 12000Current Dens ity ( A / m ² )

Ohm

ic P

olar

izat

ion

( V )

T = 1073 K

Figure 5.8: Ohmic polarization change with current density ( T = 1073 K ).

0

0,05

0,1

0,15

0,2

0,25

0,3

0 2000 4000 6000 8000 10000 12000 14000


Con

cent

ratio

n Po

lariz

atio

n ( V

)

Anode Cathode

T = 1073 K

Figure 5.9: Concentration polarization change with current density ( T = 1073 K ).

78

0

0,2

0,4

0,6

0,8

1

1,2

0 2000 4000 6000 8000 10000 12000


Cel

l Vol

tage

( V

)

0

500

1000

1500

2000

2500

3000

3500

4000

4500

Pow

er D

ensi

ty (

W /

m² )

Cell Voltage Pow er Density

T = 1073 K

Figure 5.10: Change in the cell voltage and the power density with current density ( T = 1073 K ).

79

0

0,1

0,2

0,3

0,4

0,5

020

0040

0060

0080

0010

000

1200

014

000

Cur

rent

Den

sity

( A

/ m² )

Voltage ( V )

Ano

de A

ctiv

atio

n P

olar

izat

ion

Cat

hode

Act

ivat

ion

Pol

ariz

atio

nO

hmic

Pol

ariz

atio

nA

node

Con

cent

ratio

n P

olar

izat

ion

Cat

hode

Con

cent

ratio

n P

olar

izat

ion

T =

1073

K

Figu

re 5

.11:

Cal

cula

ted

pola

rizat

ion

effe

cts w

ith c

urre

nt d

ensi

ty (

T =

1073

K ).

80

0

0,2

0,4

0,6

0,81

1,2

020

0040

0060

0080

0010

000

1200

014

000

Cur

rent

Den

sity

( A

/ m² )

Voltage ( V )

0500

1000

1500

2000

2500

3000

3500

4000

4500

Power Density ( W / m² )

Ano

de A

ctiv

atio

n P

olar

izat

ion

Cat

hode

Act

ivat

ion

Pol

ariz

atio

nO

hmic

Pol

ariz

atio

nA

node

Con

cent

ratio

n P

olar

izat

ion

Cat

hode

Con

cent

ratio

n P

olar

izat

ion

Cel

l Vol

tage

Pow

er D

ensi

ty

T =

1073

K

Figu

re 5

.12:

Cal

cula

ted

pow

er d

ensi

ty, c

ell v

olta

ge a

nd p

olar

izat

ions

with

cur

rent

den

sity

( T

= 10

73 K

).

81

82

The cathode activation polarization is obviously higher than the anode activation

polarization. This is because of the cathode’s lower exchange current density. Both the

cathode and anode activation polarizations increase almost linearly at T = 1273 K.

Ohmic polarization changes linearly with increasing current, since it is related to the

material thickness. Electrolyte section has the main effect on ohmic polarization due to

its material’s high resistivity. The effect of ohmic polarization can be changed by

changing the thickness of materials or by using another material instead.

Anode concentration polarization is higher than the cathode concentration polarization at

low current densities. But after a critical region, cathode concentration polarization

increases exponentially and goes to infinity as can be seen from the graphs. Regarding to

Figures 5.3 and 5.9, for the cell working at T = 1273 K, IL = 10788 A / m², and for the

one working at T = 1073 K, IL = 11900 A / m² are the limiting current densities.

Figures 5.4 and 5.10 give the change of the cell voltage and the power densities with

current density. The lower the temperature, the more the cell voltage drops, and the less

the power density is. This result is just as expected. In chapter 3, it was explained that

the efficiency of fuel cells decreases with increasing temperature, but on the other hand,

as it can be observed from the results, as the temperature increases, the drop in cell

voltage decreases, and power density increases.

Figures 5.5 and 5.11 show the polarization effects in a single graph for both

temperatures. The results show that at lower temperatures, the cathode activation

polarization becomes significant and has the most effect on voltage drop. The ohmic and

anode activation polarizations behave similarly at lower temperatures. The anode

concentration polarization, on the other hand, becomes significant at high temperature

values, while cathode concentration polarization has very little effect at low

temperatures and low current densities. At high temperature, cathode concentration

polarization increases exponentially. Anode and cathode activation polarizations depend

on temperature much more than the other types of polarizations. The increase in

83

activation polarization with decreasing temperature is the highest in all types of

polarizations.

Figures 5.6 and 5.12 give a better understanding of the effect of temperature on the fuel

cell voltage and the power density. For the cell operating at T = 1273 K, at I ≈ 10400 A /

m², the power density reaches its peak point P ≈ 8889 W / m². For the cell operating at T

= 1073 K, at I ≈ 8000 A / m², the power density reaches its peak point P ≈ 3955 W / m².

At I = 6000 A / m², for the cell working at T = 1273, cell voltage is, E = 0.989 V, while

for the cell working at T = 1073 K, cell voltage is, E = 0.621 V. There is a significant

difference between the two cell voltages.

As a conclusion, as the fuel cell operating temperature increases, the polarizations

decrease and this results higher cell voltages and high power densities

5.1.2 Thermodynamic Analysis

The system is assumed to be at following conditions:

• T = 1273 K., P = 1 atm.

• The air consists of 20 % O2 and 80 % N2 in molar ratios.

In order to calculate the entropies at each node of the system, the partial pressures of the

molecules must be determined. For this reason, mixture compositions at each node of the

system for ideal case ( i.e. conversion ratios are unity ) is given in Table 5.1. Mixture

compositions for different conversion ratios at each node of the system are given in

Tables 5.2 – 7.

Table 5.1: Mixture compositions for ideal case ( i.e. 14=CHη , 1

2=Hη , 1

2=Oη ).

Mole Numbers Mole Ratios Nodes Nodes 1 2 3 4 1 2 3 4

CH4 1 0 0 0 0,3125 0 0 0

H2O 2,2 0,2 0 4,2 0,6875 0,03846 0 0,31818

H2 0 4 0 0 0 0,76923 0 0

O2 0 0 2 0 0 0 0,2 0

CO2 0 1 0 1 0 0,19231 0 0,07576

N2 0 0 8 8 0 0 0,8 0,60606

Table 5.2: Mixture compositions for 9.04=CHη , 9.0

2=Hη , 9.0

2=Oη .


CH4 1 0,1 0 0,1 0,3125 0,02 0 0,00747

H2O 2,2 0,4 0 3,64 0,6875 0,08 0 0,27205

H2 0 3,6 0 0,36 0 0,72 0 0,02691

O2 0 0 2 0,38 0 0 0,2 0,02840

CO2 0 0,9 0 0,9 0 0,18 0 0,06727

N2 0 0 8 8 0 0 0,8 0,59791


2=Hη , 8.0

2=Oη .


CH4 1 0,1 0 0,1 0,3125 0,02 0 0,00746

H2O 2,2 0,4 0 3,6 0,6875 0,08 0 0,26866

H2 0 3,6 0 0,4 0 0,72 0 0,02985

84

O2 0 0 2 0,4 0 0 0,2 0,02985

CO2 0 0,9 0 0,9 0 0,18 0 0,06716

N2 0 0 8 8 0 0 0,8 0,59701

Table 5.4: Mixture compositions for , , . 9.04=CHη 9.0

2=Hη 7.0

2=Oη


CH4 1 0,1 0 0,1 0,3125 0,02 0 0,007353

H2O 2,2 0,4 0 3,2 0,6875 0,08 0 0,235294

H2 0 3,6 0 0,8 0 0,72 0 0,058824

O2 0 0 2 0,6 0 0 0,2 0,044118

CO2 0 0,9 0 0,9 0 0,18 0 0,066176

N2 0 0 8 8 0 0 0,8 0,588235


2=Hη , 8.0

2=Oη .


CH4 1 0,1 0 0,1 0,3125 0,02 0 0,007375

H2O 2,2 0,4 0 3,28 0,6875 0,08 0 0,241888

H2 0 3,6 0 0,72 0 0,72 0 0,053097

O2 0 0 2 0,56 0 0 0,2 0,041298

CO2 0 0,9 0 0,9 0 0,18 0 0,066372

N2 0 0 8 8 0 0 0,8 0,589971


2=Hη , 7.0

2=Oη .


CH4 1 0,1 0 0,1 0,3125 0,02 0 0,007353

H2O 2,2 0,4 0 3,2 0,6875 0,08 0 0,235294

H2 0 3,6 0 0,8 0 0,72 0 0,058824

O2 0 0 2 0,6 0 0,2 0,044118

85

CO2 0 0,9 0 0,9 0 0,18 0 0,066176

N2 0 0 8 8 0 0 0,8 0,588235

Table 5.7: Mixture compositions for , , . 9.04=CHη 7.0

2=Hη 7.0

2=Oη


CH4 1 0,1 0 0,1 0,3125 0,02 0 0,007278

H2O 2,2 0,4 0 2,92 0,6875 0,08 0 0,212518

H2 0 3,6 0 1,08 0 0,72 0 0,078603

O2 0 0 2 0,74 0 0,2 0,053857

CO2 0 0,9 0 0,9 0 0,18 0 0,065502

N2 0 0 8 8 0 0 0,8 0,582242

The maximum work calculated for the fuel cell unit working at T = 1273 K is, Wmax =

805.6 kJ, and for the fuel cell unit working at T = 1073 K is, Wmax = 808.8 kJ,. The

losses in work due to reformer reaction, incomplete reactions, cracks and polarizations

are calculated as a function of the conversion ratios. For each conversion ratio, the

obtainable works are calculated. Using Eq.4.15, the second law efficiencies are

calculated as functions of the conversion ratios. These calculations are repeated for each

temperature value.

For the calculations, as a first step, the pressure is assumed to be, P = 1 atm. Keeping the

pressure constant, the second law efficiencies of the fuel cell unit as functions of

different conversion ratios are calculated with different current densities. These results

are obtained for each temperature value. As a result, under constant pressure, the effects

of both the temperature and the current density on the fuel cell unit are investigated. The

results obtained are presented in Tables 5.8 – 5.15.

As a second step, in order to investigate the pressure effect on the second law efficiency

of the fuel cell unit, the calculation steps are repeated at P = 5 atm. For different current

density values, the second law efficiencies are calculated as functions of conversion

86

87

ratios, for both T = 1073 K, and T = 1273 K. The results obtained are presented in

Tables 5.16 – 5.23.

Table 5.8: Calculated second law efficiencies as functions of conversion ratios ( T = 1273 K, P = 1 atm, I = 8000 A / m² ).

methane 1 0,9 0,9 0,9 0,9 0,9 0,9

Conversion ratios hydrogen 1 0,9 0,9 0,9 0,8 0,8 0,7 oxygen 1 0,9 0,8 0,7 0,8 0,7 0,7 Maximum energy available ( kJ ) 805,6 805,6 805,6 805,6 805,6 805,6 805,6 Irreversible losses ( kJ ) due to reformer reaction 105,6 104,7 104,7 104,7 104,7 104,7 104,7 due to incomplete reactions 0 110,7 116,3 175,0 162,9 175,0 218,2 due to cracks 7 6 6 5 5 5 4 due to polarizations 133,5 108,2 106,8 93,5 96,1 93,5 84,1 Obtainable work ( kJ ) 559,445 476,001 471,737 427,390 436,820 427,390 394,538 Second law efficiency 0,694 0,591 0,586 0,531 0,542 0,531 0,490


methane 1 0,9 0,9 0,9 0,9 0,9 0,9


88


methane 1 0,9 0,9 0,9 0,9 0,9 0,9 Conversion ratios hydrogen 1 0,9 0,9 0,9 0,8 0,8 0,7

oxygen 1 0,9 0,8 0,7 0,8 0,7 0,7 Maximum energy available ( kJ ) 805,6 805,6 805,6 805,6 805,6 805,6 805,6 Irreversible losses ( kJ ) due to reformer reaction 105,6 104,7 104,7 104,7 104,7 104,7 104,7 due to incomplete reactions 0 110,7 116,3 175,0 162,9 175,0 218,2 due to cracks 7 6 6 5 5 5 4 due to polarizations 98,0 79,4 78,4 68,6 70,6 68,6 61,8 Obtainable work ( kJ ) 594,952 504,762 500,142 452,245 462,385 452,245 416,908 Second law efficiency 0,739 0,627 0,621 0,561 0,574 0,561 0,518




89







90







91


methane 1 0,9 0,9 0,9 0,9 0,9 0,9



methane 1 0,9 0,9 0,9 0,9 0,9 0,9


92


methane 1 0,9 0,9 0,9 0,9 0,9 0,9



methane 1 0,9 0,9 0,9 0,9 0,9 0,9


93







94





methane 1 0,9 0,9 0,9 0,9 0,9 0,9


For both T = 1273 K, and T = 1073 K, the change in second law efficiencies with

current densities are calculated as functions of conversion ratios. To make it clear, the

results obtained at different temperature and pressure values for 100% and 90%

conversion ratios are graphed, first by keeping temperature constant in order to

investigate the pressure effect on the second law efficiency, and secondly by keeping

pressure constant in order to investigate the effect of temperature on the second law

efficiency. The graphs are presented in Figures 5.13 – 15.6.

0,0

0,2

0,4

0,6

0,8

1,0

0 2000 4000 6000 8000 10000


Seco

nd L

aw E

ffic

ienc

y

P=1 atm, 100% conversion ratio P=1 atm, 90% conversion ratio


Figure 5.13: Second law efficiency with current density at P = 1 atm, and P = 5 atm, conversion ratios are 100% and 90% respectively, T = 1273 K.

95

0,0

0,2

0,4

0,6

0,8

1,0

0 2000 4000 6000 8000 10000


Seco

nd L

aw E

ffic

ienc

y



Figure 5.14: Second law efficiency with current density at P = 1 atm, and P = 5 atm, conversion ratios are 100% and 90% respectively, T = 1073 K.

0,0

0,2

0,4

0,6

0,8

1,0

0 2000 4000 6000 8000 10000


Seco

nd L

aw E

ffic

ienc

y

T=1273 K, 100% conversion ratio T=1273 K, 90% conversion ratio


96

Figure 5.15: Second law efficiency with current density at T = 1273 K, and T = 1073 K, conversion ratios are 100% and 90% respectively, P = 1 atm.

0,0

0,2

0,4

0,6

0,8

1,0

0 2000 4000 6000 8000 10000


Seco

nd L

aw E

ffic

ienc

y



Figure 5.16: Second law efficiency with current density at T = 1273 K, and T = 1073 K, conversion ratios are 100% and 90% respectively, P = 5 atm.

At constant temperature, it can be observed from the graphs that the second law

efficiency of the system increases with increasing pressure. The difference in second law

efficiency between a high pressure system and a low pressure one remains almost

constant ( i.e. the same ) as the current density increases.

At constant pressure, on the other hand, the second law efficiency of a low temperature

system is greater at very low current densities. But as the current density increases, the

second law efficiency of a low temperature system decreases much more than the second

law efficiency of a higher temperature system does. For current density values of

I > 4000 A / m², the second law efficiency of a high temperature system becomes much

greater than the second law efficiency of a low temperature system. Hence, if the cell is 97

98

operated at low current densities, the low temperature gives a better second law

efficiency, while for the cell operated at high current densities, the high temperature

gives a better second law efficiency.

As a conclusion, it can be stated that, for a system operating at a constant current density

value, keeping the temperature constant, the second law efficiency increases as the

pressure is increased, and keeping the pressure constant, the second law efficiency

increases as the temperature is increased, unless it is operated at very low current

density.

5.2 Results of Simulation Model 2

5.2.1 The Heat Required by The Reformer and Vaporizer

In order to determine how much fuel must be left to be burnt in the afterburner, the heat

requirement of the reformer and vaporizer must be determined, as a first step. The heat

requirement of the reformer and vaporizer reactions with percentage of methane

reforming ( or the reformer efficiency ) is presented in Figure 5.17. The fuel left at the

exit of the fuel cell stack is burnt in the afterburner, and the heat released by the

combustion process is used in reformer and vaporizer reactions. The remaining heat is

lost to the environment. The heat release by the combustion of the fuel in the afterburner

is given in Figure 5.18. The temperatures are assumed to be Tr,i = 1100 K for the

reformer reaction and T = 1200 K for the afterburner reaction, while T=298 K is the

exact temperature value for the vaporizer reaction. Tr,i is used to indicate the reformer

inlet temperature. These temperatures will differ with the calculated temperatures of the

reformer inlet and afterburner temperatures, but they are assumed so, in order to make

conservative assumptions for the reformer efficiency and the fuel utilization rate.

0

50

100

150

200

250

300

75 80 85 90 95 100

Methane Reforming ( % )

Hea

t Req

uire

d ( k

J )

Heat Required for Rem orm ing Reaction Total Heat Required Heat Required for Vaporization

Figure 5.17: Heat requirement of the components of the SOFC system with methane reforming rate ( Tr,i = 1100 K).

0

50

100

150

200

250

300

0 0,2 0,4 0,6 0,8 1 1,2

M oles of Hydrogen and M ethane Burnt

Hea

t Out

put (

kJ

)

Methane Hydrogen

99

Figure 5.18: Heat release of the combustion processes in the afterburner with respect to the mole number of the fuel that is burnt.

The reformer efficiency and fuel utilization rate directly affect the second law efficiency

of the cell. The unreacted methane and hydrogen are burnt in the afterburner as

mentioned. As the reformer efficiency and fuel utilization rate change, the heat release

due to these unreacted fuel changes as well. The comparison of the heat release of the

methane and hydrogen with decreasing reformer efficiency is given in Figure 5.19.

Following this, the change in fuel utilization rate with reformer efficiency is given in

Figure 5.20.

0

50

100

150

200

250

300

100 96 92 88 84 80 76 72

Reform er Efficiency ( % )

Hea

t Req

uire

d / R

elea

se (

kJ )

Total Heat Required Heat release by Methane Heat Release by Hydrogen

Figure 5.19: Comparison of heat release by methane and hydrogen with reformer efficiency.

100

0

0,2

0,4

0,6

0,8

1

1,2

Reformer Ef f iciency Fuel Utilization Rate

Figure 5.20: Change in fuel utilization rate with reformer efficiency.

Referring to Figures 5.19 and 5.20, the reformer efficiency of 0.9 and fuel utilization

rate of 0.75 seem to be conservative values. With a higher fuel utilization rate value,

there was not sufficient fuel left to satisfy the heating requirements in the reformer and

vaporizer reactions.

5.2.2 Heat Exchanger Design

Keeping the overall heat transfer coefficient constant for both preheaters,

U = 0.05 kW / m²K, the areas of heat exchangers are calculated according to the given

conditions in Table 5.24. The results are also given in Table 5.24.

101

Table 5.24: Heat exchanger design conditions and results.

Preheater Number

Cold Gas Stream Inlet Temperature

( K )

Cold Gas Stream Outlet Temperature

( K )

Hot Gas Stream Inlet Temperature

( K )

Hot Gas Stream Outlet Temperature

( K )

Calculated Preheater Area

( m² )

1 298 1133 1200 959 566

2 298 900 1010 560 1657

5.2.3 Thermodynamic Analysis

The 90% of methane conversion ratio ( i.e. reformer efficiency ) yields the reformer

reaction as follows.

OHCHCOHOHCH 242224 4.01.09.06.32.2 +++→+

The fuel utilization of 75% means that 75% of hydrogen will react with oxygen. Hence,

the fuel cell reaction will be,

2222 9.07.235.16.3 HOHOH +→+

The unreacted methane and hydrogen are burnt in the afterburner. Since complete

combustion is assumed, the combustion reactions are as follows.

2224 1.02.02.01.0 COOHOCH +→+

OHOH 222 9.045.09.0 →+

As seen from the reactions above, the required oxygen to sustain the system operations,

is 2 moles. Hence, 2 moles of oxygen must be supplied to the system. Since air is

assumed to be consisting of 20% O2 and 80% N2, a total of 10 moles of air is to be

supplied into the system.

102

Since the number of moles of air supplied into the system is determined, the molar

chemical compositions of the gas streams at each node can be calculated. After

determining the molar chemical compositions at each node of the system, the mass flow

rates at each node, referring to fuel supply rate of 1 kg/s, can be calculated as well. The

results are presented in table 5.25. In the table, ni and yi indicates the mole number and

molar composition of each element.

Table 5.25: Calculated molar chemical compositions of gas streams at each node. CH4 H2O O2 N2 H2 CO2 m&

Node ni yi ni yi ni yi ni yi ni yi ni yi (kg/s)

1 1 1 0 0 0 0 0 0 0 0 0 0 1

2 0 0 0 0 2 0,2 8 0,8 0 0 0 0 17,5

3 0 0 2,2 1 0 0 0 0 0 0 0 0 2,5

4 1 0,3125 2,2 0,6875 0 0 0 0 0 0 0 0 3,5

5 1 0,3125 2,2 0,6875 0 0 0 0 0 0 0 0 3,5

6 0,1 0,02 0,4 0,08 0 0 0 0 3,6 0,72 0,9 0,18 3,5

7 0 0 0 0 2 0,2 8 0,8 0 0 0 0 3,5

8 0,1 0,0074 3,1 0,2305 0,45 0,0335 8 0,5948 0,9 0,0669 0,9 0,0669 21

9 0 0 4,2 0,3182 0 0 8 0,6061 0 0 1 0,0758 21

10 0 0 4,2 0,3182 0 0 8 0,6061 0 0 1 0,0758 21

11 0 0 4,2 0,3182 0 0 8 0,6061 0 0 1 0,0758 21

To determine the electrical work output of the system, the fuel inlet and the air inlet

temperatures must be assumed. The fuel inlet temperature is assumed to be, Ta = 1000 K

and the air inlet temperature is assumed to be, Tc = 900 K. The SOFC stack is assumed

to work under I = 1000 A / m² current density.

First of all, regarding to electrochemical model, the Nernst potential is calculated. With

known anode and cathode operating temperatures, the activation and concentration

polarizations at both anode and cathode, and the ohmic polarization in the SOFC stack 103

104

are calculated. The cell voltage is obtained by subtracting the irreversibilities due to

polarizations from the Nernst potential. Hence, the electrical work output is obtained.

This net work output, i.e. the actual work, is compared with the heat input to the system

and the chemical exergy of methane to determine the first and second law efficiencies.

The results obtained are given in Table 5.26 and 5.27. Table 5.27 also gives the heat

released by the SOFC stack.

Table 5.26: Net work output of the SOFC operating at Ta = 1000 K, Tc = 900 K, I =

1000 A/m². Eo (V) ηAct,a (V) ηAct,c (V) ηConc,a (V) ηConc,c (V) ηOhm (V) EAct (V) We (kJ/s) 1,09 0,0172 0,0384 0,0039 0,0014 0,0173 1,0118 32866,456

Table 5.27: The first and second law efficiencies of model 2. We (kJ/s) Qlost (kJ/s) LHV (kJ/s) Exergy Input (kJ/s) ηI (%) ηII (%)

32866.456 2892.613 50018.703 52177.681 65.71 62.99

The enthalpy and exergy values are calculated for each node by applying the energy

balance and the exergy balance at each component. The calculated enthalpy and exergy

values and temperatures of each node are given in Figure 5.21. The comparison of input,

output and loss of energy and exergy in the system is given in Table 5.28.

105

Figure 5.21: System 2 operating at 1 atm, 90% reformer efficiency, 75% fuel utilization rate, fuel inlet temperature is 1000 K, air inlet temperature is 900 K.

SOFC

Reactor 1831.025 kJ/s

Vaporizer 2868.29 kJ/s

Preheater 2714.113 kJ/s

Mixer

325.602kJ/s

Preheater 1 357.389 kJ/s

Reformer 993.629 kJ/s

Afterburner3529.976 kJ/s

T11=556.06 K H11=5680.777 kJ/s T7=900 K 11 Ex11=4851.519 kJ/s H7=10665.35 kJ/s

Ex7=5427.977 kJ/s T2=298 K H2= 0 Ex2= 0 2 7

Wact=32866.456 kJ/s Air Inlet Qlost=2892.613 kJ/s

Ex=2101.113 kJ/s

T10=919.79 K T6=1000 K 10 H10=16346.127 kJ/s H6=65443.051 kJ/s T4=298 K T1=298 K Ex10=10993.609 kJ/s Ex6=63238.711 kJ/s H4=50018.703 kJ/s H1=50018.703 kJ/s

Ex4=53476.663 kJ/s Ex1=52177.681 kJ/s

1 4 5

6 8

3

9

T5=1034.97 K H5=55091.112 kJ/s T9=1089.07 K T8=1089.07 K Q1=10351.939 kJ/s Ex5=56712.981 kJ/s H9=21418.536 kJ/s H8=409349.332kJ/s Ex=7519.359 kJ/s

Ex9=14587.316 kJ/s Ex8=31868.094 kJ/s

Q2=6185.36 kJ/s T3=298 K H3=0 Ex=4492.874 kJ/s Ex3=1624.584 kJ/s

Qo=2393.497 kJ/s Ex=1738.569 kJ/s

Environment, To

Water

106

Table 5.28: The comparison of input, output, and loss of energy and exergy in the system. Energy and exergy values are normalized relative to the lower heating value and chemical exergy of the fuel, respectively.

Energy Exergy

Fuel Input 100 100 Work Output 65,709 62,99 Heat to the Environment 10,568 7,359 Vaporization Process 12,366 5,497 Exhaust Gas ( Node 11 ) 11,357 9,298 Irreversibility in System Units 14,856

100 100

The results obtained show that the first law and second law efficiencies for the model

simulated are 65.71% and 62.99%, respectively. The second law efficiency of the model

is lower than the first law efficiency, because of the higher chemical exergy of the fuel

than its lower heating value.

The system is modeled with the assumption that the fuel cell is supplied 100%

theoretical air. In practice, in order to maintain a relatively high oxygen concentration at

the cathode, a high air – fuel ratio is required. This high ratio of air – fuel will increase

the irreversibility.

The exhaust gas ( node 11 ) has a high temperature. This high temperature value causes

more exergy destruction. Reducing the fuel utilization rate or methane reforming rate

can decrease this high temperature value, provided that the required heat is produced in

the afterburner to sustain the system operations. The exhaust gas of the fuel cell system

being supplied with high extent of air will have a lower temperature value, since

preheater 1 in that case will destroy more exergy.

Mixer destroyed little exergy since the inlet and outlet temperatures are the same and

irreversibility is only due to mixing of methane and water vapor.

107

The most irreversible process is the combustion in the afterburner. The system with a

high fuel cell exit temperature value would have less irreversibility in the afterburner.

The irreversibility in the SOFC stack would be decreased with a low fuel cell exit

temperature value.

Vaporization process requires heat input. This irreversibility can be eliminated by

developing a recycling fuel cell system which can be used to eliminate the need for the

vaporizer.

Methane reforming causes more exergy destruction within the system. Reducing the

combustion of fuel while reforming as much methane as possible ( hence producing as

much fuel as possible ) would increase the second law efficiency.

108

CHAPTER 6

CONCLUSION

The results obtained for the simulation models show that solid oxide fuel cells have high

second law efficiencies. As discussed before, this high efficiency values are one of the

benefits of fuel cell systems. Solid oxide fuel cell’s high operating temperature caused

less voltage drop, and therefore the irreversibilities occuring inside the fuel cell were

less.

Different configurations for the system might be studied, such as assuming different

methane – water vapor ratio, supplying some extent air, or trying different reformer

efficiency and fuel utilization rate values. Results obtained from these configurations

can be compared with the results obtained in this study to give a more general discussion

to the subject.

High operating temperature of the solid oxide fuel cell system gives the advantage to be

combined with a gas turbine system for higher efficiencies. Studies on this combined

system show that high efficiency values can be obtained this way.

The exhaust gas stream of the solid oxide fuel cell system has high exergy value. This

exhaust gas may be used for local area heating.

109

The vaporizer and afterburner components of the system destroyed more exergy when

compared with the other components of the system. Hence, different alternatives should

be tried in order to increase the second law efficiency of the system.

One of the alternatives can be the elimination of vaporizer. Since water vapor is

produced at the exit of the fuel cell stack, addition of a splitter at the outlet of the fuel

cell stack might be used to recycle the required water vapor to the reformer. The

required water vapor can be recycled from the splitter to the reformer. This arrangement

would eliminate the need for the vaporizer while increasing the system’s second law

efficiency.

Another alternative should consider the heating requirement of the reformer. Since this

heat requirement is supplied by combustion of methane and hydgoren, the fuel

utilization rate and the reformer efficiency are lower. Instead of using hydrogen, some of

the methane entering the system might directly be used to heat the reformer, hence the

reformer efficiency would be higher and system would be modeled in order to reform as

much methane as possible. This way, more methane would be reformed and more fuel

would be produced. Combustion of the fuel should therefore be reduced as much as

possible. Such a model with higher reformer efficiency and fuel utilization rate would

have high second law efficiency.

These alternatives can increase the second law efficiency, and hence offers much better

results. The future works should consider these alternative configurations.

110

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111

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